BY  THE  SAME  AUTHOR 


An  Introduction  to  Laboratory  Physics 

The  subject  matter  is  similar  to  that  of  the  Theory 
of  Measurements,  but  the  treatment  is  adapted  to 
more  mature  students,  and  is  such  as  will  necessitate 
more  independent  thought. 

Cloth,  150  pp.;  5x7  X  inches;  $0.80. 

Four-Place  and  Five-Place  Tables 

Four-place  logarithms  with  tables  of  proportional 
parts;  five-place  logarithms;  four-place  natural  and 
logarithmic  circular  functions ;  squares ;  values  of 
.67449  Vn;  constants. 

Cardboard,  6  pp.;  4)^x7)^  inches;  $0.05. 

Second  Edition  in  Preparation :  Contains  also  four- 
place  squares  and  square  roots  of  all  numbers ;  exact 
squares  from  I2  to  9992;  inverse  tangents;  direct  and 
inverse  circular  measure;  density  and  specific  gravity 
of  water;  exponentials  (en  ;  logio^n  ;  e~n  ;  e~n*). 

Cardboard,  8  pp.  .  .  .  (In  press). 

Theory  of  Measurements 

Cloth,  xiv+303  pp.;  ...  $1.25. 


PUBLISHED  BY 

Dr.  Lucius  Tuttle, 

Jefferson  Medical  College, 

Philadelphia,  Pa. 


THE  THEOEY 


OF 


MEASUREMENTS 


BY 

LUCIUS  TUTTLE 

B.A.  (YALE);  M.D.  (JOHNS  HOPKINS) 

ASSOCIATE    IN    PHYSICS,  JEFFERSON    MEDICAL    COLLEGE,  PHILADELPHIA 


PHILADELPHIA 

JEFFERSON  LABORATORY  OF  PHYSICS 
1916 


o 


' 

.    •  : 


PUBLISHED    BY  THE   AUTHOR 

PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


PREFACE 

For  the  student  of  mathematics  this  book  is  intended 
to  furnish  an  introduction  to  some  of  the  applications 
of  the  exact  sciences  and  their  relation  to  the  " practical" 
sciences  and  useful  arts,  and  is  primarily  intended  to 
give  him  a  knowledge  of  facts  and  methods,  but  without 
neglecting  the  accurate  exercise  of  his  reasoning  powers. 

For  the  student  of  physical  science  it  is  intended 
especially  to  emphasize  general  considerations  of  meas- 
urement, theory  of  errors,  general  methods  of  procedure, 
quantitative  accuracy,  adjustment  of  observations,  etc., 
—topics  that  are  often  merely  mentioned  in  the  intro- 
duction or  appendix  of  a  laboratory  manual,  but  that 
need  laboratory  work  and  drill  quite  as  much  as  the 
measurements  of  the  individual  quantities  that  the  stu- 
dent will  take  up  in  his  later  work.  Where  it  is  impos- 
sible to  find  time  for  a  complete  course  of  the  kind 
described  here  it  may  be  helpful  to  use  selected  chapters 
of  the  book  as  occasion  arises,  or  the  student  may  be 
directed  to  use  it  as  a  reference  book,  or  even  to  read  it 
through  without  performing  any  of  the  experimental 
work. 

The  book  is  based  on  the  mimeographed  direction 
sheets  that  were  used  in  the  first  part  of  a  laboratory 
course  which  the  writer  gave  at  Jefferson  at  a  time 
when  there  was  no  elementary  textbook  available  that 
covered  the  required  ground.  In  addition  to  the  state- 
ments of  facts  and  theory  e.ach  of  the  chapters  of  the 
book  includes  directions  for  actual  experimental  work 
to  be  performed  by  the  student,  and  the  amount  'of  this 
work  has  been  so  planned  that  each  lesson  will  require 


377559 


vi  PREFACE 

about  the  same  length  of  time  as  any  of  the  others. 
For  the  average  student  this  will  nlean  about  three 
hours,  but  the  material  of  the  lessons  can  easily  be 
divided  into  a  greater  number  of  shorter  exercises  if 
desirable. 

With  the  more  widespread  use  of  a  laboratory  course 
of  this  sort  certain  shortcomings  of  the  author's  earlier 
" Introduction  to  Laboratory  Physics"  have  been  made 
manifest.  A  book  that  demands  more  or  less  vigorous 
mental  exercise  from  a  class  of  students  who  take  a  special 
interest  in  the  subject  will  naturally  need  more  elemen- 
tary exposition — more  detailed  statement  and  less  exer- 
citational  questioning — if  it  is  to  be  used  in  larger  classes 
where  there  is  a  greater  likelihood  of  finding  that  some 
of  the  students  are  lacking  in  interest  or  ability  or 
elementary  mathematical  training.*  Accordingly,  expla- 
nations and  directions  have  been  given  with  considerable 
detail,  partly  in  order  to  avoid  the  necessity  for  continu- 
ous oral  assistance  on  the  part  of  the  instructor,  and 
partly  to  help  the  student  to  learn  with  a  minimum  of 
deliberate  memorizing.  For  the  latter  purpose  facts 
have  occasionally  been  stated  implicitly  instead  of 
explicitly,  but,  in  such  cases,  always  with  a  later  re- 
iteration in  a  more  expositional  form.  The  course  is 
progressively  graded  in  difficulty,  with  the  object  of 
developing  the  student's  ability  as  he  proceeds  from  the 
easier  exercises  to  those  that  require  more  independent 
thought. 

There  has  been  a  certain  demand  for  the  "Introduction 
to  Laboratory  Physics"  in  connection  with  courses  of 

*  Such  detailed  directions  as  the  instructions  in  regard  to  round 
numbers,  page  119,  may  seem  superfluous,  but  they  indicate  faults 
that  have  been  found  in  the  work  of  more  than  one  of  the  students 
who  have  taken  this  course. 


PREFACE  vii 

mathematics  as  well  as  for  courses  in  physics,  and  for 
this  reason  the  requirements  of  the  mathematician  have 
been  especially  kept  in  mind  during  the  preparation  of 
the  present  book.  No  knowledge  of  trigonometry,  how- 
ever, is  presupposed,  and  none  is  imposed  upon  the 
reader  of  the  book,  the  terms  " function,"  " tangent," 
" cosine,"  etc.,  that  will  occasionally  be  found  being  used 
merely  as  convenient  abbreviations  for  ideas  that  would 
otherwise  need  a  more  cumbersome  description. 

In  the  introductory  chapter  the  commonest  mathe- 
matical deficiencies  of  the  student  are  reviewed  and  an 
opportunity  is  given  him  to  test  his  weak  points.  A 
lesson  on  logarithms  is  included,  which  can  be  omitted, 
if  preferred,  by  a  class  that  is  familiar  with  the  subject; 
but  there  are  often  members  of  such  a  class  who  cannot 
make  practical  use  of  logarithmic  tables  readily,  or  even 
accurately,  without  additional  practice,  and  to  anyone 
who  does  not  need  the  practice  it  will  not  be  at  all  irk- 
some. Care  has  been  taken  to  make  the  tables  in  the 
appendix  both  accurate  and  convenient.*  Experience 
has  shown  that  the  somewhat  unconventional  arrange- 
ment of  the  table  of  probable  errors,  page  292,  is  the 
most  satisfactory  in  actual  use.  The  table  of  logarithmic 
circular  functions  has  been  given  the  greatest  possible 
compactness.  The  columns  of  the  table  of  four-place 
logarithms  are  arranged  especially  for  the  convenience 
of  students  who  are  accustomed  to  using  scales  that  are 
subdivided  into  tenths,  and  the  proportional  parts  are 
given  in  the  same  way  as  in  the  most  carefully  constructed 
larger  tables.  None  of  the  methods  of  arranging  a  five- 
place  table  with  proportional  parts  within  the  limits  of 

*  All  of  the  tables  either  have  been  verified  from  two  independent 
sources  or  have  been  checked  by  recalculation,  and  the  proof-sheets 
have  been  revised  with  the  utmost  care. 


viii  PREFACE 

two  pages  has  ever  succeeded  in  giving  the  fifth  figure 
satisfactorily,  and  several  scientific  reference  books  have 
been  published  in  which  even  the  fourth  figure  of  such  a 
table  will  often  be  found  incorrect.  Accordingly,  for  the 
five-place  logarithms  in  the  present  volume  no  attempt 
has  been  made  to  include  proportional  parts,  but  direc- 
tions have  been  given  for  easy  interpolation  with  the  aid 
of  three-figure  logarithms. 

I  have  replaced  the  perpetually  misleading  name  for 
the  common  -representative  value  of  a  set  of  residuals 
by  one  which  does  not  have  this  objectionable  quality 
and  at  the  same  time  suggests  the  tnature  of  the  quantity 
in  question.  A  few  other  innovations  will  be  found  here 
and  there  in  the  text,  but  for  the  most  part  the  book 
follows  fairly  well-beaten  lines. 

I  have  found  it  advisable  to  devote  the  first  ten  or 
fifteen  minutes  of  the  laboratory  period  to  a  rapid  recita- 
tion based  on  the  lesson  of  the  previous  day;  and  have 
allowed  the  students  to  compare  many  of  their  important 
numerical  determinations  by  having  them  record  certain 
specified  results  each  day  upon  a  large  card  (22}^"  X 
28^/r)  that  is  kept  on  one  of  the  laboratory  tables,  and 
is  ruled  in  separate  columns  headed  by  each  student's 
name  and  having  separate  lines  for  each  datum.  For  the 
exercises  after  the  first  chapter  of  this  book  the  following 
data  may  be  suggested: 

Weights  and  measures:  density  of  a  (brass-and-air) 
weight. 

Angles:  largest  error  of  the  measured  sines. 

Significant  figures:   experimental  value  of  TT. 

Logarithms:  calculated  value  of  e. 

Small  magnitudes:  results  of  a  double  weighing. 

Slide  rule:  approximate  ratio  for  TT,  different  from  22/7. 

Graphic  representation:  temperature  at  4  p.  m. 


PREFACE  ix 

Curves  and  equations:  least  value  of  x  for  which 
exp  (—  x2)  is  indistinguishable  from  zero. 

Graphic  analysis :  equation  of  the  black  thread  experi- 
ment. 

Interpolation:  population,  according  to  second  extra- 
polation. 

Coordinates  in  three  dimensions :  altitude  at  Sixth  and 
Market  Streets. 

Accuracy :  measured  length  of  the  table,  or  its  relative 
deviation  from  the  average. 

Principle  of  coincidence:   measured  length  of  an  inch. 

Measurements:  mode  and  extremes  of  measured 
variates. 

Statistics:   average,  median,  and  quartiles  of  variates. 

Dispersion:  comparison  of  semi-interquartile  range 
with  dispersion  (10  seeds). 

Weights:  weighted  average  for  the  density  of  alumi- 
num. 

Criteria  of  rejection:  closer  values  for  the  approximate 
ratios  10  :  12  :  15. 

Least  squares:  comparison  of  black  thread  determina- 
tion (§  21)  with  least  square  determination. 

Indirect  measurements:  value  of  l/32  +  52  -f-  62  by 
geometrical  construction. 

Systematic  errors:  direction  and  amount  of  displace- 
ment of  the  second  hand. 

Most  of  the  apparatus  required  will  be  found  to  be 
included  in  that  which  is  used  in  other  physical  experi- 
ments; a  complete  list  of  what  is  needed  for  each  group 
of  two  students  is  given  here: 

2  metre  sticks  (graduated  in  tenths  of  an  inch  on  the  back). 
2  30-cm.  rulers  (graduated  in  cm.  and  mm.). 
1  50-cm3  graduated  cylinder. 
1  10-cm3  graduated  pipette. 


x  PREFACE 

1  platform  balance  or  trip  scale  with  slide  giving  tenths  of  a  gram 

(and  the  supporting  wedges  used  when  packed  for  shipment). 
1  set  of  brass  weights  (1  gm.  to  500  gm.). 
1  set  of  iron  weights  (1  oz.  to  8  oz.). 
1  pair  of  fine-pointed  dividers. 
1  pencil-compass. 
1  protractor  (provided  with  a  diagonal  scale). 

1  brass  measuring  disc  for  the  determination  of  TT. 

2  10-inch  slide  rules  that  need  not  have  celluloid  facings,  but  are 
provided  with  A,  B,  C,  D,  S,  L,  and  T  scales,  metric  equivalents, 
and  a  runner. 

1  hard  wood  block. 

2  vernier  calipers. 

100  seeds  or  other  variates. 

1  aluminum  block  for  density  measurements. 

1  set   of   "overflow   can"    and    "catch-bucket"   for   Archimedes' 
principle. 

2  square  wooden  rods  for  the  balance  pans. 

1  iron  clamp  to  hold  balance  on  cross-bar  over  table. 

1  irregular  solid  (large  wire  nail  or  strip  of  lead  that  can  be  immersed 

in  the  graduated  cylinder). 
1  small  test-tube, 
cardboard, 
string, 
fine  black  thread. 

The  student  should  have  a  watch  with  a  second  hand,* 
a  pocket-knife,  and  the  supplies  mentioned  in  the  intro- 
duction. A  clock  that  beats  audible  seconds  should 
be  available.  The  slide  rules  should  have  6745  on  the 
C  scale  marked  by  making  a  shallow  cut  with  a  sharp 
knife  and  rubbing  in  a  little  oil  pigment.  The  note- 
book (§  8)  used  at  the  Jefferson  Laboratory  of  Physics 
measures  about  eight  by  ten  and  a  half  inches  and  is 
ruled  both  horizontally  and  vertically  at  intervals  of 
one  seventh  of  an  inch. 

*  A  special  watch  for  the  laboratory,  having  a  marked  eccentricity 
of  the  second  hand,  may  be  advisable  for  the  use  of  the  students  who 
have  the  greatest  difficulty  with  the  experiment  on  periodic  errors. 


CONTENTS 


PAGE 

I.  INTRODUCTORY 1 

Object.  Purpose.  Continuity.  Results.  Fore- 
thought. Mental  Attitude.  Notes.  Material  Equip- 
ment. Mental  Equipment.  Proportionality.  Vari- 
ation. Algebraical  Formulae.  Mental  Exercises. 
Physical  Arithmetic.  Abridged  Division.  Abridged 
Multiplication.  Gradient. 

II.  WEIGHTS  AND  MEASURES 24 

C.G.S.  System.  Units  of  Length.  Units  of  Area 
and  Volume.  Units  of  Mass  and  Density.  Unit  of 
Time.  Practice  in  Using  the  C.G.S.  System.  Rule 
for  Rounding  Off  a  Half.  The  Hand  as  a  Measure. 
Measurement  of  Area.  Measurement  of  Volume. 
Measurement  of  Mass.  Measurement  of  Density. 
Equivalent  Measures. 

III.  ANGLES  AND  CIRCULAR  FUNCTIONS 37 

Unit  of  Angle.  Circular  Measure.  Numerical 
Measure  of  an  Angle.  The  Angle  ir  and  the  Unit 
Angle.  The  Protractor.  The  Diagonal  Scale. 
Measures  of  Inclination.  Use  of  a  Table  of  Tan- 
gents. Experimental  Determination  of  Sines.  Defi- 
nition of  Function.  The  Cosine  of  an  Angle. 
Circular  Functions.  Generalized  Idea  of  Angle. 

IV.  SIGNIFICANT  FIGURES 52 

Estimation  of  Tenths.  Practice.  Mistakes.  Value 
of  TT.  Physical  Measurement.  Ideal  Accuracy. 
Decimal  Accuracy.  Significant  Figures.  Relative 
Accuracy.  Calculation  of  Relative  Errors.  Rule 
for  the  Relative  Difference  of  Two  Measurements. 
Accuracy  of  a  Calculated  Result.  Accuracy  of  the 
Abridged  Methods.  Standard  Form. 

V.  LOGARITHMS 71 

Definitions.  Fundamental  Properties.  Common 
Logarithms.  Practical  Tables.  Use  of  Tables. 
The  Probability  Function. 


xii  CONTENTS 

VI.  SMALL  MAGNITUDES 80 

Approximate  Values.  Negligible  Magnitudes.  For- 
mula for  Powers.  Properties  of  Deltas.  Trans- 
formation of  Operands.  Recapitulation. 

VII.  THE  SLIDE  RULE 90 

Addition  with  Two  Scales.  Multiplication  with 
Logarithmic  Scales.  The  Slide  Rule.  Reading 
a  Logarithmic  Scale.  Multiplication.  Division. 
Ratio  and  Proportion.  Equivalent  Measures.  Re- 
ciprocals. C  and  D  Scales.  Squares  and  Square 
Roots.  Compound  Operations.  Determination  of 
Circular  Functions.  Determination  of  Logarithms 
and  Antilogarithms. 

VIII.  GKAPHIC  REPRESENTATION 102 

Indication  of  a  Point  by  Two  Numbers.  Represen- 
tation of  Two  Numbers  by  a  Point.  Representation 
of  Two  Variables  by  a  Line.  Graphic  Diagrams. 
Practice  in  Plotting  Points.  Orientation  of  a 
Graphic  Curve.  Choice  of  Scales.  General  Prin- 
ciples of  Plotting.  Representation  of  Tabular 
Values.  Smoothing  of  a  Graphic  Curve. 

IX.  CURVES  AND  EQUATIONS 115 

Graphic  Representation  of  a  Natural  Law.  Graph 
of  an  Equation.  General  Procedure.  The  Straight 
Line.  The  Parabola.  The  Probability  Curve. 
Equation  of  a  Graph.  Change  of  Scales.  Defini- 
tions of  Circular  Functions. 

X.  GRAPHIC  ANALYSIS 128 

Interpretation  of  Equations.  The  Graph  of  y  —  a -\-bx. 
The  Straight  Line  Law,.  The  "Black  Thread" 
Method.  Intercept  Form  of  a  Linear  Equation. 
The  Graph  of  y  =  a  +  bx  +  cxz.  Law  of  Density- 
Variation  of  Water.  Typical  Curves.  Linear  Rela- 
tionship by  Change  of  Variables. 

XI.  INTERPOLATION  AND  EXTRAPOLATION 142 

Definitions.  The  Principle  of  Proportionate  Changes. 
Examples  of  Linear  Interpolation.  Graphic  Inter- 
polation. Interpolation  along  a  Curve.  Insuffi- 
ciency of  Data.  Use  of  Logarithmic  Paper.  Semi- 
Logarithmic  Paper.  Extrapolation. 


CONTENTS  xiii 

XII.  COORDINATES  IN  THREE  DIMENSIONS 158 

Coordinates  of  a  Point  in  Space.  Convention  in 
Regard  to  Signs.  Loci  of  Simple  Three-Dimensional 
Equations.  Contour  Lines.  Use  of  Contour  Maps. 
Construction  of  a  Contour  Map. 

XIII.  ACCURACY 170 

Significant  Figures.  Infinite  Accuracy.  Relative 
Errors.  Uncertain  Figures.  Superfluous  Accuracy. 
Finer  Degrees  of  Accuracy.  Possible  Error  of  a 
Measurement.  Possible  Error  after  a  Calculation. 
"Probable"  Error.  Accuracy  Required  in  Special 


XIV.  THE  PRINCIPLE  OF  COINCIDENCE 180 

Effect  of  Magnitude  upon  Accuracy.  Measurement 
by  Estimation.  Measurement  by  Coincidence.  The 
Vernier.  Use  of  the  Vernier  Caliper.  Slide-Rule 
Ratios. 

XV.  MEASUREMENTS  AND  ERRORS 188 

Direct  and  Indirect  Measurements.  Independent, 
Dependent,  and  Conditioned  Measurements.  Har- 
mony and  Disagreement  of  Repeated  Measurements. 
Errors  of  Measurements.  Classification  of  Errors. 
Accidental  and  Constant  Errors.  Errors  and  Varia- 
tions. Measurement  of  Variates. 

XVI.  STATISTICAL  METHODS 197 

Frequency  Distributions.  Class  Interval.  Types  of 
Frequency  Distribution.  The  Probability  Curve. 
Representative  Magnitudes.  The  Average.  The 
Median.  The  Mode.  Choice  of  Means.  Devia- 
tions. Average  by  Symmetry.  Average  by  Parti- 
tion. Quartiles.  Semi-Interquartile  Range. 

XVII.  DEVIATION  AND  DISPERSION 211 

Characteristic  Deviations.  Total  Range.  Average 
Deviation.  Standard  Deviation.  Dispersion.  Sig- 
nificance of  the  Dispersion.  Advantage  of  the  Dis- 
persion. Calculation  of  the  Dispersion.  Rule  for 
the  Accuracy  of  the  Average.  Use  of  the  Table  of 
Dispersions.  Dispersions  with  the  Slide  Rule.  Sigma 
Notation.  Dispersion  of  an  Average.  The  State- 
ment of  a  Measurement.  Relative  Dispersion. 


xiv  CONTENTS 

XVIII.  THE  WEIGHTING  OF  OBSERVATIONS 223 

Necessity  of  Weights  for  Observations.  Density  by 
Different  Methods.  Weights  for  Repeated  Values. 
The  Weighted  Average.  Arbitrarily  Assigned 
Weights.  Weight  and  Dispersion.  Limitations  of 
w  =  k/d?.  Exception  to  the  Rule. 

XIX.  CRITERIA  OF  REJECTION 230 

Observational  Integrity.  Importance  of  Criteria. 
Chauvenet's  Criterion.  The  Probable  Error. 
Graphic  Approximation  to  Chauvenet's  Criterion. 
Irregularities  of  Small  Groups.  Justification  of  the 
1  Criterion.  Wright's  Criterion.  Comparison  of 
Characteristic  Deviations. 

XX.  LEAST  SQUARES 240 

The  Average  as  a  Least-Square  Magnitude.  Least 
Squares  for  Conditioned  Measurements.  Least 
Squares  and  Proportionality.  Least  Squares  for  a 
Theoretically  Constant  Value.  Consecutive  Equal 
Intervals.  Equal  Intervals  without  Least  Squares. 
Simultaneous  Indirect  Measurements. 

XXI.  INDIRECT  MEASUREMENTS 255 

Importance  of  Indirect  Measurements.  Probable 
Error  of  a  Sum.  Probable  Error  of  a  Difference. 
Probable  Error  of  a  Multiple.  Associative  Law. 
Probable  Error  of  a  Product.  Probable  Error  of  a 
Power.  Distributive  Law.  Recapitulation.  Graphs 
of  Propagated  Errors.  Relative  Importance  of  Com- 
pounded Errors. 

XXII.  SYSTEMATIC  AND  CONSTANT  ERRORS 265 

Definitions.  Test  for  Systematic  Errors.  Example 
of  a  Systematic  Error.  Example  of  a  Periodic  Error. 
Example  of  a  Progressive  Error.  Constant  Errors. 

APPENDIX 275 

Explanatory  Note.  Formulae.  Equivalents. 
Greek  Alphabet.  Size  of  Errors.  Characteristic  De- 
viations. General  Sources  of  Error.  Density  of 
Water.  Inverse  Tangents  and  Circular  Measure. 
Squares  and  Square  Roots.  The  Probability  In- 
tegral. Five-Place  Logarithms.  Exponentials. 
Fifth  Place  of  Logarithms.  Four-Place  Logarithms. 
Squares.  Constants.  Circular  Functions. 

INDEX .  293 


I.  INTRODUCTORY 

1.  Object. — The  object  of  a  course  in  the  theory  of 
measurements  is  not  only  to  give  a  certain  knowledge  of 
the  scientific  facts  that  are  studied,  but  also  to  develop 
the  thinking  and  reasoning  powers  and  to  furnish  the 
special  kind  of  mental  training  that  results,  in  the  first 
place,  from  practice  in  making  various  kinds  of  measure- 
ments with  particular  care  for  their  accuracy,  and,  in 
the  second  place,  from  the  consideration  of  accuracy  in 
its  quantitative  aspects, — from  realizing  that  accuracy 
itself  can  be  made  a  subject  of  measurement,  that  there 
are  relative  degrees  of  accuracy,  that  accuracy  is  im- 
portant in  one  place  and  means  only  a  waste  of  effort 
in  another,  that  absolute  accuracy  is  an  impossibility, 
that  a  measurement  by  itself  is  of  much  less  value  thai* 
when  accompanied  by  a  statement  of  its  precision. 

When  a  course  of  physical  measurements  is  used  as  a 
preliminary  to  laboratory  work  or  practical  work  in  some 
subject  such  as  astronomy,  physics,  psychology,  or 
surveying  it  is  further  of  value  in  giving  the  student  a 
certain  familiarity  with  apparatus  and  a  facility  in 
handling  it  in  such  a  manner  that  he  acquires  the  habit 
of  utilizing  it  to  the  best  advantage  and  of  keeping  it  in 
such  condition  that  it  is  most  fully  utilizable  when 
needed. 

2.  Purpose. — It  is  advisable  for  the  student  to  have  it 
pointed  out  at  the  beginning  of  the  course  that  his  con- 
scious purpose,  throughout  all  of  his  work,  should  be  to 
learn,  rather  than  to  accomplish  the  assigned  exercise.     In 
order  to  help  him  in  his  education  and  training  he  is 
permitted  to  do  certain  laboratory  work  which  will  make 

2  1 


2  l  THEORY- OF  .MEASUREMENTS  §5 

his  learning  easier  and  its  effects  more  lasting  and  more 
useful  to  him.  He  should  tell  himself  that  the  work  is 
allowed,  rather  than  that  it  is  required,  in  case  he  has 
any  tendency  to  look  upon  his  course  of  study  as  a 
tedious  job,  and  thus  make  it  irksome. 

3.  Continuity. — In  any  graded  course  of  study,  where 
each  exercise  is  an  advance  beyond  the  previous  ones  and 
requires  a  knowledge  of  the  earlier  work,  it  is  a  decided 
handicap  for  the  student  to  miss  any  of  the  work.     If 
an  absence  cannot  be  avoided  he  should  take  particular 
pains,  for  his  own  sake,  to  make  up  the  work  by  outside 
study.     This  is  especially  important  for  any  study  that 
is  at  all  mathematical  in  character.     See  that  each  topic 
is  thoroughly  comprehended  before  going  on  to  the  next. 
The  work  will  usually  be  easier  if  each  lesson  is  read  over 
before  coming  into  class,  so  that  it  is  not  necessary  to 
begin  the  class-room  work  as  a  new  and  unfamiliar  sub- 
ject. 

4.  Results.  — In  order   to   obtain   good   results   it   is 
necessary  for  the  student  to  preserve  an  attitude  of  alert 
attention  toward  his  own  work,  and  especially  not  to  omit 
any  part  of  it  or  postpone  it.     Be  thorough  without 
being  in  haste;  better  to  have  half  of  the  day's  work  done 
and  done  well  than  to  try  to  take  in  all  of  it  without 
having  any  of  it  more  than  half  assimilated. 

5.  Forethought. — Whenever   any   piece   of   apparatus 
is  used  it  must  be  kept  in  mind  that  it  may  be  needed 
again  later,  and  it  is  as  important  to  keep  it  in  good  order 
as  it  is  to  use  it  efficiently.     It  does  not  take  long  for  the 
consequences  to  show  in  the  student's  work  if  he  is 
accustomed  to  pick  up  an  instrument  where  he  happens 
to  find  it,  and  drop  it  as  soon  as  its  immediate  need  has 
passed.     A  certain  orderliness  in  the  handling  of  ap- 
paratus is  a  habit  that  is  well  worth  cultivating,  as  much 


I  INTRODUCTORY  3 

for  its  effect  on  the  student's  individuality  as  for  the  con- 
servation of  property  values. 

6.  Mental  Attitude. — In  addition  to  keeping  in  mind 
the  fact  that  it  is  much  more  important  to  learn  principles 
than  to  "work  through"  each  day's  lesson  the  student 
should  adopt  the  motto  that  in  all  kinds  of  studying  it  is 
better  to  think  than  to  memorize.     For  some  students  it 
seems  only  too  easy  to  get  into  the  habit  of  concentrating 
upon  individual  items  and  memorizing  isolated  state- 
ments of  fact  without  ever  understanding  their  bearings 
or  realizing  their  inter-relationships  or  acquiring  a  larger 
comprehension  of  the  body  of  scientific  knowledge  which 
is  built  up  on  them.     The  best  help  to  a  broader  vision 
lies  in  thinking  over  the  facts  that  one  comes  across. 
Just  as  important  as  the  question  "  What  is  true?"  is  the 
further  question  "Why  is  it  true?"     Better  than  a  brain 
packed  full  of  facts  is  a  mind  that  can  reason  out  what 
the  facts  must  necessarily  be  in  particular  cases.  Memory 
with  little  reasoning  power  is  useless  for  any  highly 
organized  living  being;  but  reasoning  power  with  little 
memory  would  be  perfectly  practicable,  as  long  as  such 
things  as  paper  and  pencil  can  be  had.     Furthermore, 
the  ability  to  think  for  oneself  is  of  the  greatest  utility  in 
enabling  the  student  to  rely  upon  his  own  observational 
powers.     The  untrained  student  is  prone  to  ask  "Is  my 
result  right?"  in  circumstances  where  the  student  who 
has  learned  to  stand  on  his  own  feet  knows  that  no  one 
has  a  better  knowledge  of  what  the  "right"  result  is  than 
himself. 

7.  Notes. — For  any  practical  work,  or  laborator}^  work, 
it  is  important  that  the  student's  notes  shall  be  accurate 
and  that  they  shall  be  complete.     Neatness  is  usually 
worth  while,  but  it  is  distinctly  secondary  in  importance 
to  thoroughness  and  accuracy.     Time  spent  on  beauti- 


4  THEORY  OF  MEASUREMENTS  §7 

fying  the  notebook  by  means  of  elaborately  shaded 
drawings  or  painstaking  arrangement  of  matter  is  usually 
time  wasted.  All  notes  should  be  clear  and  intelligible, 
and  in  such  shape  that  they  can  be  readily  understood  by 
anyone  who  has  an  ordinary  knowledge  of  the  subject 
that  they  deal  with.  Each  day's  work  is  always  to  be 
dated,  and  it  is  advisable  to  write  a  heading  in  such 
a  way  that  it  will  catch  the  eye  at  once,  and  show  at  a 
glance  where  one  day's  work  ends  and  the  next  begins, 
as  well  as  indicating  the  nature  of  the  following  matter 
after  the  manner  of  a  title.  A  certain  amount  of  "  dis- 
play," by  underlining  or  otherwise  should  also  be  given 
to  two  other  things,  the  statement  of  original  measure- 
ments, especially  when  further  calculations  depend  upon 
them,  and  the  final  results  of  such  calculations. 

The  matter  of  making  on3's  notes  thorough  needs  little 
explanation;  they  should  be  made  a  digest  of  all  that  the 
student  learns  and  does  in  the  laboratory  course,  but 
without  copying  or  duplicating  matter  in  his  textbook 
that  he  can  easily  turn  to  when  it  is  wanted.  Any 
questions  that  are  asked  in  the  text  should  be  answered 
in  the  notebook. 

The  matter  of  accuracy  is  one  that  requires  some  care 
and  alertness.  It  is  necessary  to  make  it  a  rule  that  all 
work  done  with  a  pen  or  pencil  must  be  done  in  the  note- 
book and  everything  in  the  notebook  must  be  put  down 
consecutively,  in  its  natural  order.  If  the  student  uses 
the  last  pages  of  the  notebook  for  miscellaneous  calcu- 
lations it  is  usually  impossible  to  find  a  particular  piece 
of  work  when  it  happens  to  be  wanted  at  some  later 
time.  Under  no  circumstances  is  any  work  with  pen  or 
pencil  to  be  done  on  scraps  of  paper,  or  in  a  "  temporary 
notebook"  or  anywhere  else  except  in  its  proper  place 
and  order  in  the  permanent  notebook.  The  slight  gain 


I  INTRODUCTORY  5 

in  neatness  of  the  student's  notes,  which  is  usually  the 
object  of  such  procedures,  is  not  nearly  important  enough 
to  counterbalance  the  possibility  of  errors  in  copying 
data  and  the  probability  of  being  later  obliged  to  hunt 
in  vain  for  statements  that  are  not  in  their  proper  place. 
The  notebook  of  the  scientist,  like  that  of  the  accountant, 
should  be  a  book  of  "  original  entry,"  and  for  reasons 
that  are  as  important  for  a  scientific  investigation  of 
natural  phenomena  as  they  are  for  a  legal  investigation 
of  indebtedness.  Furthermore,  no  measurement  that 
has  been  written  down  in  the  notebook  should  ever  be 
rubbed  out  with  an  eraser;  if  there  is  a  reason  for  doubting 
its  value,  or  even  if  it  is  obviously  wrong  it  may  be  can- 
celed by  drawing  a  line  through  it,  but  this  should  be 
done  in  such  a  way  as  not  to  obscure  what  is  written 
down  but  to  permit  it  to  be  utilized  later  if  it  is  found 
desirable  to  do  so. 

8.  Material     Equipment. — The     student's     notebook 
should  be  of  such  size  and  character  as  will  be  best 
adapted  to  his  work.     If  it  is  not  furnished  by  the 
Department  directions  will  be  given  in  regard  to  the  kind 
of  notebook  that  should  be  used.     Pen  and  ink  will  be 
needed,  for  notes  that  are  taken  with  a  pencil  are  almost 
always   unsatisfactory.     A   fountain   pen   is   advisable, 
although  not  necessary.     A  piece  of  blotting  paper  should 
be  obtained  which  is  long  enough  to  reach  across  the 
page  of  the  notebook.     A  hard  pencil  with  the  point  kept 
well  sharpened  will  also  be  needed. 

9.  Mental  Equipment. — The  student  of  the  theory  of 
measurement  should  have  had  a  good  course  in  algebra 
as  far  as  the  solution  of  equations  of  the  first  degree;  also 
a  sufficient  knowledge  of  plane  geometry  to  include  the 
properties  of  perpendiculars,   equal  triangles,   isosceles 
and  similar  triangles,  the  theorem  of  Pythagoras,  and 


6  THEORY  OF  MEASUREMENTS  §10 

the  properties  of  similar  figures.  Certain  arithmetical 
processes,  such  as  the  method  of  extracting  square  root 
or  cube  root,  are  not  needed  for  purposes  of  physical 
measurement,  but  a  good  grasp  of  certain  others,  such 
as  proportion  and  variation,  is  almost  a  necessity.  An 
intelligent  comprehension  of  principles  is  as  important  as 
a  memory  of  rules  and  formulae. 

10.  Proportionality. — The  rule  that  the  product  of  the 
means  is  equal  to  the  product  of  the  extremes  is  one  that 
can  be  used  for  the  solution  of  almost  any  problem  in  pro- 
portionality, but  the  important,  thing  for  the  student  is  to 
understand  the  meaning  of  the  combination  of  terms  that 
constitutes  what  is  called  a  " proportion.'7  For  example, 
the  value  of  a  commodity  is  proportional  to  its  amount; 
thus,  it  might  be  that  13  Ibs.  :  26  Ibs.  ::  43c.  :  •  •  -c. 
The  student  who  can  handle  such  an  example  only  by 
multiplying  26  by  43  and  then  dividing  by  13  is  wasting 
much  of  his  time  on  mathematical  drudgery  that  might 
be  much  more  profitably  devoted  to  the  study  of  other 
subjects.  A  proportion  is  by  definition  an  equality  of 
ratios;  the  ratio  of  13  Ibs.  to  26  Ibs.  is  stated  to  equal  the 
ratio  of  43c.  to  some  other  number  of  cents,  and  a 
glance  will  show  that  the  second  weight  is  twice  as  large 
as  the  first,  whence  the  second  cost  is  twice  43c.,  namely 
86c.  A  ratio  is  the  same  as  a  fraction  or  a  quotient,  and 
13  Ibs.  :  26  Ibs.  means  13  lbs./26  Ibs.  or  13  Ibs.  -r-  26  Ibs., 
or  1/2.  It  is  equally  correct  to  state  that  the  ratio  of 
quantity  to  cost  of  a  commodity  is  constant  (cceteris 
paribus),  so  that  the  above  proportion  may  just  as  well 
be  written  in  the  form  13  Ibs.  :  43c.  ::  26  Ibs.  :  86c. 
Here  the  first  ratio  is  not  an  abstract  number  like  1/2, 
but  is  a  certain  number  (about  .302)  of  pounds  for  a 
cent;  and  the  second  ratio,  which  is  stated  to  be  equal  to 
it  (::  means  the  same  as  =),  must  also  be  a  number  of 


I  INTRODUCTORY  7 

pounds  for  a  cent.  Some  students  may  find  it  advisable 
to  write  every  proportion  that  they  use  in  the  fractional 
form 

13  Ibs.  _  26  Ibs.  13  Ibs.  =  43c 

~43cT  x  26  Ibs.  ~~x   ' 

In  either  of  these  the  possibility  of  dividing  by  13  is  seen 
at  once,  and  in  this  form  no  uncertainty  can  be  felt  in 
regard  to  what  should  be  done  with  proportions  like 

13  :  43  ::  26  :  x  ::  39  :y  ::  6.5  :  z* 

Furthermore,  there  can  be  no  difficulty  in  answering 
questions  such  as  the  following,  which  sometimes  make 
trouble  for  the  student  who  fails  to  realize  the  meaning 
of  proportionality:  "The  pressure  of  a  gas  is  stated  to  be 
proportional  to  its  temperature.  How  can  these  two 
things  be  proportional  if  it  always  takes  four  terms  to 
make  a  proportion?" 

The  equality  of  two  or  more  ratios,  which  is  the  essen- 
tial of  proportionality,  is  often  expressed  by  the  use  of 
some  symbol  to  denote  a  constant  or  invariable  quantity. 
Thus  13  Ibs.  :  43c.  =  .302  lb./c.;  26  Ibs.  :  84c.  =  .302 
lb./c.;  6.5  Ibs.  :  21. 5c.  =  .302  lb./c.,  and  in  general  for 
this  particular  commodity  any  weight  -r-  corresponding 
value  =  .302  lb./c.  Following  the  usual  custom  of  using 
the  letter  c  or  k  to  denote  a  constant  it  might  be  said  of 
commodities  in  general  that  w/v  =  k.  From  this  it 
follows  that  v/w  is  equal  to  l/k,  which  is  another  constant, 

*  In  case  any  uncertainty  is  felt  the  student  should  attack  it  at 
once,  and  should  not  be  satisfied  until  the  difficulty  has  been  success- 
fully overcome.  It  is  perhaps  hardly  necessary  to  point  out  the 
fact  that  a  mathematical  subject  cannot  usually  be  read  as  fluently 
as  a  novel.  To  have  each  letter  and  symbol  observed  by  the  eye, 
or  even  read  aloud,  is  not  enough  unless  the  mind  is  given  time  for  a 
thorough  comprehension  of  the  meaning. 


8  THEORY  OF  MEASUREMENTS  §11 

say  c,  that  is  usually  called  "  price."  Since  I/. 302  =  3. 31 
the  price  of  the  substance  considered  above  must  be  3.31 
cents  per  pound,  which  is  a  constant  for  that  commodity, 
but  of  course  the  constant  c  (cost  divided  by  weight)  will 
have  a  value  different  from  3.31c./lb.  if  some  other  sub- 
stance is  considered.  In  technical  language  such  a 
" variable  constant"  is  called  a  parameter. 

Is  there  any  difference  between  what  is  stated  by 

f  Xi  =  cyi} 
\  xz  =  cyz, 

and  what  is  stated  by  x\  :  yi  ::  Xz  :  2/2?     Prove  it. 

If  the  absolute  temperature  and  the  pressure  of  a 
given  portion  of  gas  are  proportional  what  will  happen 
to  its  pressure  if  the  gas  has  its  absolute  temperature 
doubled  or  tripled? 

11.  Variation. — If  the  price  of  a  commodity  remains 
constantly  3.31  c./lb.  the  value  is  said  to  vary  in  accord- 
ance with  the  weight,  or,  shortly,  to  vary  as  the  weight. 
or,  more  explicitly,  to  vary  directly  as  the  weight.  Here 
the  weight  is  considered  to  be  a  variable  quantity,  that 
is,  we  may  consider  any  weight  we  please,  the  weight  of 
the  substance  may  assume  any  numerical  value  for  the 
purposes  of  the  discussion.  Under  these  circumstances 
the  cost  will  also  vary.  Doubling  the  weight  will  double 
the  cost;  cutting  the  weight  in  half  will  reduce  the  cdst 
by  50  per  cent;  etc.  When  any  change  in  one  quantity 
that  can  vary  is  always  accompanied  by  an  equal  relative 
chaige  in  a  second  quantity  the  variables  are  said  to  be 
proportional,  or  to  be  directly  proportional,  and  each  is 
said  to  vary  directly  as  the  other  one. 

If  a  portion  of  a  gas  is  subjected  to  compression  it  will 
be  found  that  doubling  the  pressure  exerted  upon  it  will 
cause  its  volume  to  decrease  to  only  one  half  of  its  former 


I  INTRODUCTORY  9 

amount;  multiplying  the  pressure  by  five  will  reduce  the 
volume  to  one  fifth;  etc.  This  kind  of  variation  is  called 
inverse,  and  the  pressure  and  volume  are  said  to  be 
inversely  proportional;  the  volume  is  said  to  vary  in- 
versely as  the  pressure.  Suppose  that  the  volume  is  6 
quarts  when  the  pressure  is  one  atmosphere;  then  if  the 
pressure  is  raised  to  3  atmospheres  the  volume  will  be 
reduced  to  2  quarts,  but  if  it  is  diminished  to  1/2  at- 
mosphere the  gas  will  expand  enough  to  occupy  a  space  of 
12  quarts.  If  we  write  an  equation 

v  =  k  - 
p 

it  will  be  evident  that  any  increase  in  the  size  of  the  de- 
nominator, p,  will  cause  a  relatively  equal  decrease  in  the 
size  of  the  term,  v;  and  we  have  already  seen  that  this 
equation  means  the  same  as 

vi  :  I/pi  ::  vz  :  l/pz  '•'-  v9  :  I/PS 
Clearing  the  equation  of  fractions  gives 

pv  =  k 

and  the  ordinary  way  of  expressing  the  fact  that  two 
variables,  such  as  p  and  v,  are  inversely  proportional  is  to 
write  down  an  equation  in  which  their  product  is  stated 
to  be  equal  to  a  constant;  just  as  direct  proportionality  is 
expressed  by  making  their  quotient  equal  to  a  constant. 
It  is  perhaps  worth  noticing  here  that  there  may  easily 
be  other  forms  of  variation,  in  which  there  is  no  propor- 
tionality at  all.  The  distance  traveled  by  a  train  is  not 
usually  strictly  proportional  to  the  number  of  hours  that 
elapse  during  the  process,  nor  is  an  individual's  wealth 
proportional  to  his  age.  The  matter  of  irregular  vari- 
ation will  be  taken  up  later. 


10  THEORY  OF  MEASUREMENTS  §13 

12.  Algebraical  Formulae.  —  The  following  are  some  of 
the  facts  of  algebra  which  experience  has  shown  that 
students  of  the  theory  of  measurement  need  but  are  not 
in  every  case  familiar  with: 

Letters  are  used  for  generalized  numbers  ;  if  (a  +  6)  (a  —  b) 
=  a2-62  then  (20  +  1)  X(20-l)  =  202-12,  and  similarly 
for  any  other  numbers.  —  "  Terms,"  between  plus  or 
minus  signs,  are  to  be  evaluated  before  performing  the 
additions  or  subtractions;  thus  2+4X3  —  1+4(3  —  1)  is 
equal  to  21,  not  to  31  or  any  other  number.  —  The  product 
a3  X  a4  is  a7,  not  a12;  this  is  obvious  if  it  is  written  or  con- 
sidered as  (a  X  a  X  a)  X  (a  X  a  X  a  X  a).  —  The  prod- 
uct of  a  negative  and  a  positive  number  is  negative,  but 
of  two  negatives  is  positive;  e.  g.,  (a  +  b  '  —  c)(x  —  y) 
=  ax  —  ay  +  bx  —  by  —  ex  +  cy.  —  A  negative  exponent 
indicates  a  reciprocal;  a~2  means  I/a2.  —  A  fractional 
exponent  indicates  a  root;  a*  means  V  a;  x*  =  -Six* 
=  (-\/x)3.  —  Fractional  radicands  may  be  simplified  as 
shown  in  the  following  example: 


~ 
6  V 


—The  binomial  theorem  is 
(a  +  x)n  =  an  +  na^x  + 


(a  +  b)2  =  a2  +  2ab  +  62; 
(m  -  n)3  =  ra3  -  3m*n  +  3mn2  -  n3;  etc. 

13.  Mental  Exercises.  —  The  following  list  of  exercises 
covers  various  subjects  that  students  have  occasionally 
been  found  lacking  in  familiarity  with,  also  many  that 


I  INTRODUCTORY  11 

will  be  needed  in  different  parts  of  the  course.  The 
student  should  read  each  question  and  decide  upon  the 
answer  mentally  and  without  hesitation.  If  the  answer 
is  instantly  apparent  mark  the  question  with  a  check  ( \/ ) 
and  take  up  the  next  one,  but  if  it  occasions  any  hesitation 
or  uncertainty  mark  it  with  a  plus  sign  (+)  and  if  it 
cannot  be  answered  at  all  inside  of  a  few  seconds  mark 
it  with  a  zero  (0).  Go  through  the  whole  list  rapidly, 
and  then  ask  the  advice  of  the  instructor  in  regard  to  it. 
This  will  often  make  a  great  difference  in  the  ease  of 
performing  the  later  laboratory  work.  Remember  that 
it  is  not  an  examination  of  how  much  it  is  possible  to 
recall  from  the  depths  of  your  memory,  but  a  test  of 
how  much  mathematics  you  have  in  an  immediately 
available  condition. 

1.  What  is  the  square  of  x  —  a? 

2.  What  is  the  value  of  (m  +  x)(m  —  x)l 

3.  State  the  value  of  23. 

4.  What  is  the  numerical  value  of  TT? 

5.  Which  is  the  larger  37/147  or  38/148?     Do  you 
know  of  any  general  method  of  deciding  such  a  question? 

Write  the  following  in  the  form  of  decimal  fractions: 

6.  1/3. 

7.  4/5. 

8.  1/7. 

9.  1/8. 

10.  2/9. 

11.  1/11. 

12.  Can  the  constant  TT  be  called  a  parameter?     Why? 

13.  Reduce  1/25  to  hundredths  mentally. 

14.  Is  6  twice  as  large  as  4?     How  many  times  as  large? 

15.  What  is  the  fourth  term  of  1000  :  100  ::  31  :  .  .  .? 

16.  State  the  value  of  1/500  as  a  decimal  fraction. 

17.  Simplify  the  following:  (a7)5;  a7  +  a5;  a7  X  a5. 


12  THEORY  OF  MEASUREMENTS  §13 

18.  State  the  cube  of  a  +  6. 

19.  (100  -  12)  X  (100  +  12)  =  -  -  -? 

20.  If  the  circumference  of  a  circle  is  15  cm.  in  what 
way  can  the  diameter  be  expressed? 

21.  Twenty  inches  on  a  certain  map  represents  2,000 
miles.     What  is  its  scale  of  miles  per  inch  numerically 
equal  to? 

22.  State  "one  out  of  every  four"  as  a  percentage. 

23.  What  does  "twenty  percent"  mean? 

24.  Reduce  0.375  to  a  percentage. 

25.  What  is  the  reciprocal  of  2/7? 

26.  What  percent  is  the  number  .005  equal  to? 

27.  What  percent  is  .00072? 

28.  2.84  X  10-4  =  •••? 

29.  284  X  10-4  «?'•>;•? 

30.  Find  (1  +  I)4  by  the  binomial  theorem. 

31.  Solve  3  :  12  ::  16  :  x. 

32.  How  much  is    (-  2)(-  15)/(-  5)? 

33.  What  is  the  value  of  1  +  J  +  f  +  J  +  TV  +  •  •  •  ? 

34.  Solve  mentally  42  :  x2  ::  252  :  752. 

35.  Is 

a  — 


true  for  numerical  values?     Give  an  example  of  it. 

36.  If  pv  is  a  constant  how  will  v  be  affected  by  doubling 
p? 

37.  3  +  (4  -  4  -f-  2)  (7  X  4  -  3)  =  ? 

38.  l  +  l-2  +  l-2-3  +  l-2-3-4  =  ? 

39.  State  the  approximate  value  of  8.5  -r-  10. 

40.  Does  52  X  92  equal  452?     Use  algebraical  letters 
to  illustrate  the  general  principle  that  is  involved. 

41.  State  the  approximate  square  root  of  each  of  the 
following:  2560  (ans.:  about  50);  256;  25.6;  2.56;  0.256. 

42.  What  is  the  approximate  value  of  .01/2.38? 


INTRODUCTORY 


13 


43.  Write  "49.78  thousandths  of  a  centimetre"  in  the 
form  n  cm.,  where  n  is  a  decimal  fraction. 

44.  Does 

13.29  X  0.81 
3826/41 

have  a  value  of  about  5,  or  about  50,  or  about  500? 

45.  In  the  equation 

what  is  the  value  of  y  if  x  is  zero? 

46.  In  the  equation  y  =  2x  +  4  what  is  the  value  of  y 
when  x  =  -  3? 

47.  Complete  the  following  sentence  in  the  obvious 
way:  A  cubic  inch  of  lead  weighs  165  gm.,  and  is  150  gm. 
heavier  than   1   cubic  inch   of  water;   therefore  .  .  .  . 

48.  Substitute  1/ra  mentally  for   —  log  y  in  the  ex- 
pression l/(—  log  y)  and  simplify. 

49.  Write  the  value_of  .0011  X  .00011. 

50.  Solve  p  =  2irVllg  for  I,  and  then  for  g. 

51.  If 

i 

21 


n  =  —. ,  A  /— 
sr 


how  does  n  change  if  I  becomes  f  of  its  former  size? 
t  becomes  f  as  large  as  before?     If  s  is  J  as  large? 
52.  Fill  out  the  following  table: 


If 


n 

n* 

2"  -3 

l/n 

0 
1 
2 
3 
2x 

14.  Physical  Arithmetic. — Before  performing  any  writ- 
ten calculation  with  numbers  that  have  been  obtained 


14  THEORY  OF  MEASUREMENTS  §14 

from  physical  measurements  it  is  advisable  to  make  a 
rough  mental  calculation  of  the  approximate  value  of  the 
final  result.  For  example,  at  3.31c.  per  Ib.  what  will 
5J  Ibs.  cost?  One  of  these  factors  is  a  little  more  than 
three;  the  other  is  somewhat  less  than  six.  Their 
product,  accordingly,  must  be  in  the  neighborhood  of 
18.  A  train  travels  155.8  miles  in  2|  hours;  what  is  its 
rate  in  miles  per  hour?  Here  the  time  is  less  than  3 
hours,  so  the  speed  must  be  greater  than  155.8  -f-  3;  and 
still  greater  than  150  -f-  3.  Ans. :  Somewhat  faster  than 
50  miles  per  hour.  If  a  still  closer  approximation  should 
be  desired  it  could  be  obtained  by  noticing  that  the 
actual  distance  is  about  4  percent  greater  than  150,  and 
the  actual  time  is  one  twelfth  (say  6  percent)  less  than 
the  assumed  time.  Increasing  50  by  4  percent  and  then 
by  6  percent  would  make  it  about  10  percent  larger, 
giving  55  miles  per  hour  for  a  closer  value  of  the  speed. 
The  arithmetically  accurate  value  is  56.65454545. 
Is 

4.71  X  13.8 
9.06 

equal  to  about  7,  or  about  70,  or  about  700? 

What  is  the  approximate  value  of  7.26  X  .0328? 

Point  off  00000928800000  so  as  to  make  it  equal  to 
the  product  of  .0216  and  .0043 

Point  off  the  right-hand  side  of  the  equation  2./.3 
=  00666. 

Reduce  the  ratios  in  the  following  expressions  to  ap- 
proximate percentages,  performing  the  calculation  men- 
tally: "8  Ibs.  in  every  23  Ibs.  of  sea  water  is  solid  salt" 
(ans. :  8  in  24  would  be  33 1  percent;  8/23  is  a  little  greater 
and  must  be  34  or  35  percent;  or:  8/23  =  16/46,  16/46 
>  15/45,  /.  8/23  >  1/3);  " seven  inhabitants  out  of  every 


I  INTRODUCTORY  15 

38  are  voters"  (ans.:  7  out  of  35  would  be  20  percent, 
7/42  =  16f  percent,  7/38  must  have  some  intermediate 
value,  say  18  percent);  " f ourteen-carat  gold  is  14/24 
pure"  (ans.:  14/24  =  28/48  =  56/96;  this  fraction  has  its 
numerator  about  half  as  large  as  its  denominator  and  so 
will  not  be  much  changed  by  adding  4  to  the  latter  if  2 
is  added  to  the  former,  56/96  =  58/100  =  58  percent); 
"boiling  water  will  dissolve  .000022  of  its  weight  of 
silver  chloride"  (ans.:  .000022  =  .0022  percent);  "a 
saturated  salt  solution  has  a  strength  of  5  :  13"  (ans.: 
a  little  less  than  5  :  12J  or  10  :  25  or  40  :  100,  say  38 
percent);  "a  steep  railroad  grade  may  have  a  rise  of  as 
much  as  180  feet  per  mile"  (ans.:  180  ft.  per  5280  ft.  is 
less  than  180  per  5,000  or  360  per  10,000  or  36/1000  or 
.036  or  3.6  percent). 

Notice  the  different  expressions  that  .are  in  common  use 
to  denote  the  comparison  between  a  definite  fractional 
part  and  the  total.  The  same  meaning  is  expressed  by 
each  of  the  following  phrases  as  by  any  of  the  other  ones : 
22  per  million,  22  out  of  a  million,  22  in  1000000, 
22/1000000,  .000022,  22  :  1000000,  and  .0022  per  centum 
or  .0022  percent. 

Observe  also  that  the  fractional  notation  is  more  con- 
venient than  the  word  per  in  naming  compound  units  of 
measurement,  and  has  the  same  significance.  Thus,  a 

speed  of  40  miles  per  hour  is  customarily  written  40  T— -, 

or  40  mi/hr,  meaning  40  times  1  mile  per  hour  or  40  -r-r . 

That  this  notation  is  consistent  will  be  made  obvious  by 
considering  that  40  X  1  mile  X  1  hour  would  necessarily 
be  the  same  as  40  X  1  mile  X  60  minutes,  which  reduces 
to  2400  X  1  mile  X  1  minute,  and  40  miles  per  hour  is 
quite  different  from  2400  miles  per  minute;  but  40  X  1 


16  THEORY  OF  MEASUREMENTS  §15 

mile  -h  1  hour  =  40  X  1  mile  -r-  60  minutes  =  f  X  1 
mile  -T-  1  minute,  and  f  of  a  mile  per  minute  is  plainly 
the  same  speed  as  40  miles  per  hour.  Of  course  there 
are  some  kinds  of  compound  units  which  are  properly 
expressed  when  their  component  simple  units  are  multi- 
plied together  instead  of  being  divided.  For  example, 
one  foot-pound  of  energy  is  equal  to  16  foot-ounces,  a 
fact  that  could  not  be  true  if  the  unit  were  1  X  1  foot 
-r-  1  pound,  but  that  requires  it  to  be  1  X  1  foot  X  1 
pound.  (Insulation  resistance  in  "ohms  per  mile"  and 
lighting  efficiency  in  "  watts  per  candle"  furnish  illus- 
trations of  mis-named  units.  The  longer  wire  has  the 
lesser  insulation  resistance  so  that  20  ohms  for  one  mile 
is  the  same  as  10  for  two  miles  and  the  only  rational  name 
for  the  unit  is  the  ohm-mile.  Efficiency  means  light  per 
energy  and  its  unit  would  properly  be  candle-power  per 
watt,  but  the  illuminating  engineer  prefers  to  consider 
inefficiency,  which  is  properly  measured  by  watts  per 
candle-power.) 

15.  Abridged  Division.— A  number  that  is  obtained  as 
the  result  of  a  physical  measurement  is  frequently  needed 
for  some  kind  of  a  calculation.  When  this  is  the  case  it 
is  a  fact  (as  will  be  shown  later)  that  the  final  result  never 
needs  to  be  expressed  with  a  greater  number  of  figures 
than  the  original  data  contained.  Thus,  it  may  be 
possible  to  measure  the  width  of  a  table  so  carefully  as 
to  make  sure  that  the  measurement  is  62  centimetres 
+  3  millimetres  +  8  tenths  of  a  millimetre.  Such  a 
quantity  is  preferably  written  as  a  number  of  centimetres, 
and  in  this  case  is  62.38,  a  number  consisting  of  four 
figures.  Suppose  it  is  necessary  to  find  out  what  one 
third  of  the  width  will  amount  to.  One  third  of  62.38  is 
20.793333 .  .  . ,  and,  as  stated  above,  four  figures  of  this 
result,  namely  20  79,  are  all  that  are  necessary.  As  a 


I 


INTRODUCTORY 


17 


275)1558(566545 
1375 
1830 
1650 
1800 
1650 
1500 
1375 


matter  of  fact,  to  keep  more  than  four  figures  would  be 
decidedly  objectionable.  If  the  original  measurement 
gave  the  correct  number  of  tenths  and  hundredths  of  a 
centimetre  without  pretending  to  state  any  knowledge  of 
the  correct  number  of  thousandths  how  could  any  cal- 
culation assume  to  give  correct  figures  in  thousandth's 
and  tens-of-thousandth's  places?  Similarly,  in  §  14, 
the  " arithmetically  accurate"  value 
would  be  wrong  if  the  given  distance 
were  even  a  thousandth  of  an  inch 
longer  or  if  the  time  varied  from  an 
exact  2f  hours  by  as  much  as  a  mill- 
ionth of  a  second.  Here  one  of  the 
numbers  (155.8  miles)  has  four  figures 
while  the  other  can  hardly  be  consid- 
ered to  have  more  than  three  (2.75 
hours).  In  such  cases  it  is  a  fact 
that  the  final  result  will  have  only  as 
many  trustworthy  figures  -as  there  are 
in  the  shortest  number  from  which  it  is 
derived.  When  a  number  having  four 
figures  is  divided  by  a  number  of  three 
figures  there  should  be  only  three  fig- 
ures kept  in  the  quotient. 

The  principle  just  stated  makes  it 
possible  to  employ  the  " abridged" 
processes  of  multiplication  and  divi- 
sion, which  will  automatically  give 
just  the  right  number  of  figures  in  the 
answer,  and  will  also  save  considerable 
labor  on  the  part  of  the  computer. 

The  first  example  shown  in  the  mar- 
gin has  been  worked  out  by  ordinary 
"long"   division;  in  the  second  one 
3 


1100 
1500 
1375 
125 


2^)1558(567 

1375 

183 

165 

18 

I9 

ABRIDGED  DIVI- 
SION. —  After  each 
subtraction  the  next 
step  is  to  shorten 
the  divisor  instead 
of  to  lengthen  the 
dividend. 


18  THEORY  OF  MEASUREMENTS  §15 

the  abridged  method  has  been  used.  The  latter  process 
differs  from  the  former  in  only  one  respect:  Whenever 
the  process  of  " bringing  down"  a  zero  would  be  employed 
the  last  figure  of  the  divisor  is  canceled  instead.  In 
order  that  the  temporary  dividend  shall  be  larger  than 
the  divisor  one  method  stretches  out  the  dividend  by 
affixing  a  cipher;  the  other  shortens  the  divisor  by 
trimming  off  its  last  figure.  A  comparison  of  the  two 
examples  will  show  that  the  same  result  is  achieved  in 
each  case.  The  beginner  should  work  out  the  quotient 
of  the  two  numbers  given  above,  canceling  the  last 
figure  of  the  divisor  whenever  he  would  otherwise 
"bring  down"  a  zero,  but  not  referring  to  the  illustration 
until  he  was  finished:  After  the  first  subtraction,  when 
the  divisor,  275,  is  not  contained  in  the  remainder  the 
divisor  is  shortened  by  crossing  off  the  final  5.  Then, 
27  is  contained  in  183  six  times.  The  next  step  is  to 
multiply  27  by  6.  Before  saying  6  X  7  =  42  notice 
that  if  the  5  had  not  been  canceled  there  would  have 
been  3  to  carry,  resulting  in  45  instead  of  42.  Accord- 
ingly 5  is  written  down  instead  of  2  and  the  rest  of 
the  multiplication  proceeds  as  usual.  After  the  next 
subtraction  gives  a  remainder  of  18  the  divisor  is  short- 
ened to  2  instead  of  having  the  new  dividend  lengthened 
to  180.  Then,  2  would  be  contained  in  18  just  9 
times,  but  considering  the  figure  that  was  last  canceled 
it  is  plain  that  2.7  will  not  be  contained  much  more  than 
6  times.  Six  times  the  canceled  7  would  be  42  and  would 
give  4  to  carry,  so  6  times  the  2  (plus  4)  is  written  16. 
The  next  subtraction  and  cancellation  puts  an  end  to  the 
work.  In  the  specimen  given  above  it  has  been  noticed 
that  the  third  figure  of  the  quotient  is  to  be  the  last  one, 
and  it  has  been  written  down  as  a  7  instead  of  a  6  because 
the  next  product,  19,  comes  nearer  in  value  to  the  re- 


INTRODUCTORY 


19 


quired  18  than  would  the  number  16.  In  other  words  the 
quotient  is  nearer  567  than  566  and  so  the  larger  number 
represents  it  more  accurately  than  the  smaller. 

The  quotient  is  to  be  pointed  off  by  making  a  pre- 
liminary mental  calculation,  as  explained  in  §  14.     For 
example  if  the  original  numbers  had  been  2.75  and  155.8 
the  answer  would  have  been  56.7. 
Similarly,    1558./.0275  =  56700; 
1.558/27.5  =  .0567;  etc. 

Divide  180.000  by  3.1416 
without  referring  to  the  work 
given  in  the  margin  until  the 
answer  has  been  obtained.  The 
sixth  figure  which  is  given  in  the 
quotient  is  intended  to  represent 
the  value  of  the  2/3  remaining 
after  the  last  subtraction.  It 
could  also  have  been  obtained 
by  continuing  the  regular  process 
of  abridged  division:  cancel  the 
3,  leaving  0.$  for  the  divisor; 
then  2  -=-  0.3  =  7;  multiplying, 
7  X  ?  gives  2  to  carry,  7X0 
+  2  =  2. 

Find  the  quotient  if  the  divisor  is  236453  and  the  divi- 
dend is  6764309.  The  answer  should  come  out  28.60741. 
After  the  figure  4  of  the  quotient  has  been  written  down 
the  next  step  is  to  multiply  230^^0  by  4.  Ordinarily  it 
is  sufficient  to  take  the  nearest  canceled  figure  and  say 
4  X  6  =  24,  giving  2  to  carry;  but  as  the  product 
comes  close  to  25,  which  is  on  the  boundary  between  2 
to  carry  and  3  to  carry,  it  is  well  to  investigate  one  more 
canceled  figure,  saying  4  X  £  =  16,  giving  2  to  carry 
toward  4  X  0;  the  latter  then  becomes  26,  giving  3  to 


157080 

22920 

21991 

929 

628 

301 

283* 

18 

16 

2 

ABRIDGED  DIVISION.  — 
At  the  mark  *  notice  that 
3.6  comes  nearer  to  being 
"  4  to  carry  "  than  "  3  to 
carry,"  since  io  is  more 
than  three  and  a  half. 


20  THEORY  OF  MEASUREMENTS  §16 

carry  toward  the  written  product  instead  of  2.  Make 
up  and  work  out  two  examples  in  which  a  long  number  is 
divided  by  a  short  number,  and  vice  versa.  Follow  the 
regular  routine:  divide,  multiply,  subtract,  and  cancel; 
and  repeat  as  many  times  as  necessary. 

16.  Abridged  Multiplication. — The  trustworthiness  of 
a  product  of  two  or  more  numbers  follows  the  same  rule 
as  that  .of  a  quotient :  no  more  figures  of  the  product  are 
"  significant ','  than  the  number  of  them  which  the  shortest 
factor  contains.  Thus  if  the  diameter  of  a  circle  is 
found  to  be  8  centimetres  +  0  millimetres  +  0  tenths 
of  a  millimetre  but  nothing  is  stated  about  hundredths  of 
a  millimetre,  so  that  only  three  figures,  8.00,  of  the 
diameter  are  known,  it  will  not  be  possible  to  obtain 
more  than  three  figures  of  the  circumference,  even  if 
the  other  factor,  3.14159265359,  contains  a  dozen  figures. 
Here  too  the  ordinary  arithmetical  process  can  be 
abridged  so  as  to  save  time  and  work,  and,  what  is  more 
important,  to  avoid  being  misled  by  figures  that  have 
been  kept  when  they  should  have  been  discarded. 

The  illustration  (a)  shows  the  ordinary  process.  It  is 
just  as  easy,  although  not  customary,  to  use  the  figures  of 

(a)   65.97       (6)   65.97          (c)  60.#f 

24.13           24.13  24.JL3 

19791  13194  13194 

6597            26388  2639 

26388              6597  66 

13194               19791  20 

1591.8561  1591.8561  1591.9 

GENESIS  OF  THE  ABRIDGED  METHOD. — (a)  Ordinary  long  multi- 
plication. (6)  The  same  with  the  figures  of  the  multiplier  used  in 
the  reverse  order,  (c)  The  same  as  (6),  but  the  partial  products  are 
kept  from  stringing  out  to  the  right  by  progressively  shortening  the 
multiplicand. 


1  INTRODUCTORY  21 

the  multiplier  in  the  reverse  order,  multiplying  first  by 
2,  then  by  4,  1,  and  3,  and  " stepping"  the  partial  prod- 
ucts successively  one  more  place  to  the  right  instead  of 
to  the  left.  This  has  been  done  in  illustration  (6). 
Examine  it  closely,  and  see  that  you  understand  just 
how  the  process  (6)  is  carried  out  and  why  it  must 
necessarily  give  the  same  result  as  (a). 

The  abridged  method  is  shown  at  (c).  The  first 
multiplication  is  by  the  left-hand  figure  2  as  in  (6). 
Then  the  last  figure  of  the  multiplicand,  7,  is  canceled; 
and  the  next  multiplication  (by  4)  is  begun  directly  under 
the  first.  The  process  of  canceling  and  multiplying  is 
continued  in  the  same  way  until  either  the  multiplicand 
is  entirely  canceled  or  the  multiplier  has  been  entirely 
utilized,  and  the  result  is  pointed  off  in  accordance  with 
the  directions  in  §  14.  The  student  should  work  out  the 
same  example  independently,  remembering  to  investigate 
how  much  there  is  "to  carry"  from  the  figure  last 
canceled.  In  the  last  partial  product  of  (c)  the  3X6 
=  18  is  increased  by  two  units  because  3  X  $  gives  just 
1.5  to  carry  but  it  is  evident  that  3  X  ffl  must  give  a 
number  nearer  to  2  than  to  1. 

Multiply  the  quotient  28.60741  given  above  by  the 
divisor  236453,  and  notice  that  the  dividend  is  found  cor- 
rectly as  far  as  six  figures  (more  than  676430^),  which  is 
all  that  can  be  expected  if  one  factor  contains  only  six 
figures. 

The  value  of  180/Tr  is  57.29578.  Multiply  this  by 
3.141593  and  see  if  you  obtain  180.0000  correct  to  seven 
figures. 

Multiply  any  number  that  has  five  figures  by  some 
number  having  only  two  figures.  Repeat  the  multi- 
plication, using  the  shorter  number  for  multiplicand  and 
the  longer  one  for  multiplier.  Which  method  Is  pref- 


22  THEORY  OF  MEASUREMENTS  §17 

erable,  in  view  of  the  statement  at  the  beginning  of  §  16? 

17.  Gradient. — If  a  road  that  goes  up-hill  rises  2  feet 
for  every  5  feet  of  horizontal  distance  it  is  said  to  have  a 
slope  of  2  :  5,  or  2/5,  or  0.4,  or  2  in  5.  That  is,  the  ratio 
of  any  vertical  rise  to  the  corresponding  horizontal 
distance  is  taken  as  a  numerical  measure  of  its  steepness. 
Of  course  the  slope  could  also  be  measured  in  degrees; 
in  the  case  just  mentioned  the  "grade  angle,"  or  angle 
which  the  slant  line  makes  with  a  horizontal  line,  would 
be  twenty-two  degrees — too  steep  to  be  satisfactory  for  a 
road-way, — but  the  usual  custom  is  to  state  the  measure 
of  a  slope  in  terms  of  vertical  rise  per  horizontal  distance. 
Another  way  of  looking  at  the  same  thing  is  to  consider 
it  as  the  amount  of  rise  per  unit  of  horizontal  distance; 
thus,  a  road  that  rises  two  feet  in  every  five  will  of  course 
have  a  rise  of  2/5  of  a  foot  for  a  single  foot  of  horizontal 
distance. 

A  level  road  is  one  that  has  no  slope.  That  is,  its 
slope,  when  measured  as  rise  per  level  distance,  amounts 
to  zero.  If  a  line  is  made  to  slant  more  and  more  steeply 
the  ratio  that  represents  its  slope  will  obviously  become 
greater  and  greater  without  limit;  thus,  a  slope  of  1000 
would  be  hardly  distinguishable  from  a  true  vertical, 
and  yet  between  these  two  there  must  be,  for  example,  a 
slope  of  1000000000.  The  diagonal  of  a  square  is  inclined 
to  any  of  the  sides  at  an  angle  of  45°,  and  its  slope  is  of 
course  unity.  These  facts  may  be  abbreviated  as  follows : 
slope  of  0°  =  0;  slope  of  45°  =  1;  slope  of  90°  =  QO. 

Draw  roughly  an  equilateral  triangle  that  has  one  side 
horizontal.  Draw  a  vertical  from  its  apex  to  the  middle  of 
its  base,  and  prove  that  the  gradient  of  a  sixty-degree 
slope  is  equal  to  V  3 ;  also  that  if  the  angle  is  half  as 
large  as  this  the  slope  will  be  only  one  third  as  much. 

It  is  sometimes  convenient  to  consider  the  steepest 


I  INTRODUCTORY  23 

possible  " slope"  (a  vertical  line)  as  having  100  percent 
of  steepness.  This  will  be  the  case  if  we  measure  the 
amount  of  slant  not  by  the  ratio  of  rise  to  level  distance 
but  by  the  ratio  of  rise  to  slant  distance.  A  road  which 
has  a  grade  angle  of  22°  will  rise  nearly  3  feet  for  every 
8  feet  of  distance  along  its  slanting  surface,  and  this 
measure  of  steepness  may  be  called  percent  slope  to 
distinguish  it  from  the  slope,  or  gradient,  as  previously 
defined. 

Prove  that  the  "percent  slope"  of  45°  is  H  2;  of  0° 
is  zero;  of  90°  is  1;  of  30°  is  |  (use  the  same  bisected  equi- 
lateral triangle). 

The  steepest  slopes  that  are  generally  used  for  road- 
ways range  from  12  percent  to  15  percent.  On  good 
turnpikes  the  grades  are  almost  always  kept  below  3 
percent.  Two  percent  is  decidedly  steep  for  a  railroad 
grade,  and  in  modern  good  railroad  construction  one 
percent  is  about  the  maximum.  For  slight  inclinations, 
such  as  railroad  grades,  the  difference  between  rise  per 
horizontal  distance  and  rise  per  slant  distance  is  neg- 
ligibly small.  Thus,  for  1°  they  are  respectively  .017455 
and  .017452,  or  92.16  feet  per  mile  and  92.15  feet  per 
mile. 


II.     WEIGHTS  AND   MEASURES 

Apparatus. — Ruler;  metre  stick;  graduated  cylinder; 
graduated  pipette;  pair  of  dividers;  irregular  solid; 
platform  balance;  set  of  gram  weights;  set  of  ounce 
weights;  towel;  glass  jar  or  "  catch-bucket ";  small  test 
tube. 

18.  C.G.S.  System. — The  older  units  of  measurement, 
such  as  the  length  of  a  barleycorn,  the  width  of  a  man's 
palm,  or  the  length  of  a  foot  or  a  pace,  were  objectionable 
chiefly  on  account  of  their  lack  of  uniformity.  Not  only 
did  different  countries  use  Different  units  for  measuring 
quantities  of  the  same  kind,  but  even  when  a  unit  of  the 
same  name  was  used  in  different  localities  its  value  was 
not  the  same.  In  most  civilized  countries  these  older 
units  have  been  entirely  superseded  by  a  new  system  of 
weights  and  measures,  and  in  all  countries  this  system 
has  come  into  universal  use  for  every  kind  of  scientific 
work.  It  is  usually  called  the  C.G.S.  System,  from  the 
initial  letters  of  the  units  of  length  (the  centimetre),  of 
mass  (the  gram),  and  of  time  (the  second).  These  three 
units  are  called  fundamental,  because  they  have  been 
arbitrarily  fixed  in  size,  while  all  the  other  units  of  the 
system  have  been  so  chosen  as  to  make  them  depend 
upon  these  three  in  as  simple  a  manner  as  possible. 
For  example,  the  derived  unit  of  velocity  is  such  as  will 
denote  movement  through  a  single  centimetre  of  distance 
in  a  single  second  of  time,  thus  making  the  measure  of  a 
velocity  numerically  equal  to  the  quotient  of  space 
1  traversed  divided  by  time  elapsed  during  the  process. 
Similarly,  the  density  of  an  object  is  defined  as  its  mass 
in  grams  divided  by  its  volume  in  cubic  centimetres,  so 

24 


II 


WEIGHTS  AND  MEASURES 


25 


that,  although  no  name  has  been  given  to  the  unit,  the 
density  which  is  numerically  equal  to  unity  must  be  the 
density  of  such  a  substance  as  will  weigh  one  gram  for 
each  cubic  centimetre  of  its  volume;  the  unit  of  force  is 
the  force  that  must  act  for  one  second  of  time  in  order 
to  produce  a  change  of  one  unit  (one  centimetre  per 
second)  in  the  velocity  of  a  unit  mass. 

19.  Unit  of  Length. — The  scientific  unit  of  length  is 
the  centimetre.  It  is  equal  to  about  half  a  finger-breadth 
and  is  often  found  on  tape-measures, 
rulers,  etc.  These  are  simply  copies 
of  accurate  standards  belonging  to 
the  manufacturer,  which  in  turn  owe 
their  accuracy  to  a  careful  comparison 
with  the  standards  of  the  government. 
In  the  case  of  the  governments  that 
subscribed  to  the  Metric  Convention, 
including  the  United  States,  the  stand- 
ards, which  are  called  national  proto- 
type metres,  are  lengths  of  one  metre 
(i.  e.,  100  centimetres)  carefully  laid 
off  between  lines  near  the  ends  of 
certain  bars  of  artificially  aged  plat- 
inum-iridium  alloy  which  are  102  cen- 
timetres long  and  have  a  cross-section 
that  somewhat  resembles  a  letter  X 
(Fig.  1).  The  greater  length  is  used 
instead  of  a  single  centimetre  because 
it  can  be  measured  more  accurately, 
and  the  cross-section  is  for  the  purpose 
of  giving  rigidity,  and  in  order  to  allow 
the  scale  to  be  marked  on  a  surface 
that  would  be  neither  stretched  nor 
compressed  if  the  bar  should  be  slightly  bent.  The 


FIG.  1.  TRESCA 
CROSS-SECTION. — A 
bar  of  this  shape  has 
great  rigidity  for  a 
given  weight  of 
metal;  and  a  slight 
bending,  such  as 
would  stretch  the 
lower  part  and  com- 
press the  upper 
part,  has  no  effect 
on  the  "  neutral 
web,"  n,  where  the 
scale  is  engraved. 
The  diagram  shows 
the  exact  size  of  the 
cross-section  of  the 
prototype  metres. 


26  THEORY  OF  MEASUREMENTS  §19 

standards  were  constructed  at  Paris  and  distributed  by 
lot  among  the  signatories  to  the  Metric  Convention  about 
1889,  after  being  carefully  tested  and  compared  with  one 
another  so  that  their  relative  errors  and  equations  were 
accurately  known.*  One  of  these,  which  is  kept  at  the 
International  Bureau  of  Weights  and  Measures,  near 
Paris,  is  known  as  the  international  prototype  metre  and 
corresponds  in  length  with  the  original  flat  platinum  bar 
(100  cm.  X  0.4  cm.  X  2.5  cm.)  constructed  for  the 
French  Government  by  Borda  and  called  the  metre 
des  archives.  The  Borda  standard  was  intended  to  equal 
one  ten-millionth  of  the  length  of  the  meridian  quadrant 
passing  through  Paris  from  the  north  pole  to  the  equator, 
the  earth  itself  thus  furnishing  the  original  standard. 
The  metal  bar,  however,  is  now  taken  as  the  fundamental 
standard,  not  only  because  a  microscopic  measurement  of 
it  can  be  made  more  easily  and  more  accurately  than  a 
geodetic  survey,  but  also  because  the  actual  length  of 
the  earth's  quadrant  is  not  constant.  Its  average  length, 
according  to  the  best  estimations,  is  about  10,002,100 
metres. 

The  multiples  and  subdivisions  of  the  metre  that  are  in 
actual  use  are  the  kilometre  (1  km.  is  1000  metres,  or 
about  5/8  of  a  mile;  closer  approximations  are  given  in 
the  appendix),  the  centimetre  (1  cm.),  the  millimetre 
(0.1  cm.),  and  the  micron  (0.0001  cm.).  These  are  more 
convenient  than  a  single  unit  in  some  cases,  but  in  scien- 
tific work  it  is  desirable  to  express  all  lengths  in  terms  of 

*  Modern  processes  of  measurement  are  so  accurate  that  a  dif- 
ference can  generally  be  found  between  two  standards  that  were 
intended  to  be  equal,  no  matter  how  carefully  they  were  constructed. 
The  "  equation  "  of  a  prototype  metre  expresses  the  way  in  which 
its  length  varies  when  its  temperature  is  changed.  Thus  at  any 
ordinary  temperature,  t,  the  length  in  centimetres  of  prototype 
metre  No.  18  is  99.9999  +  .0008642^  +  .000000100J2. 


II  WEIGHTS  AND  MEASURES  27 

the  accepted  unit,  1  cm.,  in  order  to  avoid  possible  con- 
fusion, or  serious  error  in  case  the  denomination  of  a 
quantity  should  be  accidentally  omitted.  Thus,  the 
length  of  the  earth's  quadrant  is  1,000,210,000,  but  in 
order  to  be  doubly  safe  it  is  advisable  to  make  it  a  rule 
to  write  the  denomination  after  a  number  in  all  cases 
(1,000,210,000  cm.). 

20.  Units  of  Area  and  Volume — The  scientific  unit 
of  area  is  the  square  centimetre  (1    cm2),  the  area  of  a 
square  each  of  whose  sides  is  1  cm.  in  length.     A  square 
foot  is  about   1000  cm2.      The   unit  of  volume  is  the 
cubic  centimetre   (1    cm3),  the  volume  of  a  cube   that 
measures  1  cm.  on  each  edge.     The  dry  and  liquid  quarts 
are  each  approximately  equal  to  1000  cm3. 

21.  Units  of  Mass  and  Density. — The    scientific    unit 
of  mass,  or  for  practical  purposes  the  unit  of  weight  in 
vacuo  at  sea  level,  latitude  45°,  is  the  gram  (1  gm.), 
which  is  divided  into  1000  milligrams  (mgm.)  just  as  the 
metre  is  divided  into  1000  millimetres.     It  is  derived 
from  kilogram  prototype  standards  (1  kgm.  =  1000  gm. 
=  2.2  Ibs.)  established  at  the  same  time  as  the  standards 
of  length  and  was  originally  intended  to  be  equal  to  the 
mass  of  one  cubic  centimetre  of  water  under  standard 
conditions.     More  careful  measurements,  however,  on 
water  that  has  been  freed  from  dissolved  air  have  shown 
that  even  at  the  temperature  of  its  greatest  density 
(3.98°  C.)  a  gram  of  water  occupies  a  trifle  more  space 
than  one  cubic  centimetre,  although  the  excess  is  only 
one  sixtieth  as  great  as  it  is  at  ordinary  room  temperature. 
In  cases  where  the  slight  change  of  volume  that  is  pro- 
duced by  heating  or  cooling  can  be  neglected  it  may  be 
considered  that  water  has  a  density  equal  to  one,  density 
being  defined  as  mass  in  grams  divided  by  volume  in 
cubic  centimetres. 


28  THEORY  OF  MEASUREMENTS  §23 

22.  Unit  of  Time.— The  scientific  unit  of  time  is  the 
second,  which  is  the  1/86400  part  of  the  length  of  an 
average  day  from  noon  to  noon.     As  the  length  of  the 
solar  day  varies  at  different  seasons  of  the  year  the  second 
is  determined  in  practice  as  1/86164.1  of  the  time  of  a 
complete  rotation  of  the  earth  with  respect  to  the  fixed 
stars.     This  unit  of  time  was  in  use  before  the  adoption 
of  the  C.G.S.  System  and  is  familiar  to  every  one.     It  is 
perhaps  worth  noticing  that  fairly  accurate  seconds  can 
be  counted  off  by  repeating,  at  ordinary  conversational 
speed,  "one  thousand  and  one,  one  thousand  and  two, 
one  thousand  and  three,"  etc. 

23.  Practice  in  Using  the  C.G.S.  System. — The  scien- 
tific system  of  units  is  so  largely  used  that  the  student 
should  not  be  satisfied  with  the  mere  ability  to  translate 
measurements    from    one    system    to    the    other.     He 
should    practice    first    estimating    (guessing)    and    then 
measuring  the  dimensions  of  various  objects  that  he 
comes  across,  until  he  has  acquired  a  certain  ability  to 
"think"  in  centimetres,  cubic  centimetres,  grams,  etc., 
instead  of  in  inches,  quarts,  and  pounds. 

Across  the  top  of  one  page  of  the  notebook  draw  a 
horizontal  line  just  ten  centimetres  long  and  rule  two 
short  perpendicular  lines  across  its  ends  in  order  to 
indicate  the  exact  length  clearly.  In  the  same  way,  along 
the  right-hand  edge  of  the  page,  draw  a  line  twenty-five 
centimetres  in  length.  Under  the  first  line  draw  five 
others  of  various  lengths  without  measuring  them.  After 
they  have  been  drawn  measure  each  one  with  a  scale  of 
centimetres  and  millimetres,  and  record  its  length  to  the 
nearest  millimetre.  For  example,  if  the  length  appears 
to  be  about  152f  mm.  it  should  be  recorded  as  15.2  cm.; 
if  about  152f  mm.  it  should  be  called  15.3  cm.  Write 
each  length  as  a  number  of  centimetres,  not  153  mm.  nor 


II  WEIGHTS  AND  MEASURES  29 

15  cm.  3  mm.,  and  make  it  a  rule  to  see  that  the  denomi- 
nation of  a  measurement  is  never  omitted. 

24.  Rule  for   "  Rounding   Off  "   One  Half.— It  may 
happen  that  the  measured  length  is  so  near  152J  mm. 
that  it  is  impossible  to  decide  between  152  and  153.     A 
fraction  perceptibly  less  than  a  half  should  be  discarded 
and  more  than  a  half  should  always  be  considered  as 
one  more  unit,  but  when  it  is  uncertain  which  figure  is 
the  nearer  one  the  universally  adopted  rule  is  to  record 
the  nearest  even  number  rather  than  the  odd  number  that 
is  equally  near.     The  reason  for  this  procedure  is  that 
in  a  series  of  several  measurements  of  the  same  quantity 
it  will  be  as  apt  to  make  a  record  too  large  as  it  will  to 
make  one  too  small,  and  so  in  the  average  of  several  such 
values  will  cause  but  a  slight  error,  if  any.     If  the  rule 
were  that  the  half  should  be  always  increased  to  the  next 
larger  unit  the  errors  would  not  balance  one  another  and 
the  average  would  tend  to  be  brought  up  to  a  larger 
value  than  it  should  have.     The  same  advantage  would 
of  course  be  obtained  if  the  nearest  odd  number  were 
always  used,  but  the  even  number  has  one  slight  addi- 
tional merit,  namely,  that  in  case  it  should  have  to  be 
divided  by  two  a  recurrence  of  the  same  situation  would 
be  avoided. 

By  comparison  with  the  lines  already  drawn  make  a 
mental  estimate  of  the  length  and  width  of  the  note- 
book; then  verify  the  estimate  by  measuring  with  a 
scale.  Remember  to  record  clearly  all  the  experimental 
work  that  is  done;  thus,  the  completed  notes  should 
show  at  a  glance  which  number  is  the  actual  width  and 
which  is  the  rough  estimate. 

25.  The  Hand  as  a  Measure. — Lay  your  hand  across  a 
ruler  or  a  metre  stick  and  either  spread  the  fingers  slightly 
or  crowd  them  closer  together,  as  may  be  necessary,  so 


30 


THEORY  OF  MEASUREMENTS 


as  to  make  a  whole  number  of  finger-breadths  occupy  the 
same  amount  of  space  as  a  whole  number  of  centimetres. 
Then  hold  the  hand  in  a  similar  manner  while  using  it  for 
practicing  approximate  measurements  of  various  objects. 
Record  also  the  exact  measurements  of  the  same  objects 
as  they  are  obtained  later  with  the  graduated  scale. 

Separate  the  thumb  and  little  finger  as  far  as  can  be 
conveniently  done  without  special  effort  and  measure 
your  span  in  centimetres  and  millimetres.  Repeat  this 
measurement  five  times,  being  careful  not  to  let  the  sight 
of  the  scale  under  your  hand  influence  the  extent  to  which 
the  fingers  are  spread,  and  decide  which  is  the  most 
satisfactory  value.  Measure  the  length  and  the  breadth 
of  the  table  by  means  of  successive  spans,  using  hand- 
breadths  or  finger-breadths  for  the  final  fraction  of  a 
span,  and  compare  the  result  with  the  actual  length  and 
breadth. 

26.  Measurement  of  Area. — Ask  the  instructor  to 
draw  an  irregular  outline  in  your  notebook  (Fig.  2). 


FIG.  2.  IRREGULAR  AREA. — The  simplest  method  of  measuring 
an  area  marked  on  squared  paper  is  to  count  the  squares  that  are 
entirely  within  the  figure  as  units  and  those  that  are  cut  by  the 
boundary  as  half  units.  The  sum  gives  the  total  area  almost  as 
well  as  if  an  attempt  were  made  to  estimate  the  fractional  size  of 
each  cut  square. 


II  WEIGHTS  AND  MEASURES  31 

Count  the  number  of  the  small  squares  of  the  ruled  paper 
which  are  entirely  included  within  it,  but  do  not  outline 
them  on  the  diagram.  To  their  total  add  half  the  number 
of  the  squares  that  are  cut  by  the  boundary  of  the  figure. 
The  result  will  be  the  area  of  the  irregular  outline,  not 
in  square  centimetres,  of  course,  but  in  terms  of  the  small 
ruled  squares,  and  its  denomination  may  be  written 
"  n  's."  Try  to  obtain  a  more  accurate  value  for  the 
total  area  by  estimating  as  closely  as  possible  how  many 
tenths  of  each  cut  square  is  included  within  the  boundary 
and  adding  these  actual  fractions.  The  first  result 
should  agree  quite  closely  with  this  one  because  any  one 
of  the  fractions  of  a  square  is  as  likely  to  be  less  than 
one  half  as  it  is  to  be  more  than  one  half,  and  the  most 
probable  value  for  the  average  of  the  fractions  is  just 
one  half. 

As  an  alternative  method,  block  off  an  equal  area  on 
the  same  irregular  figure  by  drawing  several  rectangles 
and  triangles  to  cover  it  (Fig.  3).  If  one  corner  of  a 


FIG.  3.  AREA  BY  MENSURATION. — An  irregular  figure  can  be 
"  blocked  out  "  by  a  number  of  triangles  and  parallelograms  which 
are  so  drawn  that  an  error  of  excess  in  one  place  is  approximately 
balanced  by  an  error  of  defect  in  another,  so  as  to  make  the  com- 
bined areas  of  the  geometrical  figures  equal  to  the  required  area. 
The  geometrical  areas  are  then  evaluated  by  ordinary  mensuration. 


32 


THEORY  OF  MEASUREMENTS 


§28 


triangle  projects  considerably  beyond  the  irregular  line 
see  that  one  of  its  sides  is  drawn  so  as  to  include  less  than 
the  requisite  amount  and  try  to  make  the  two  opposite 
errors  balance  as  nearly  as  possible.  Then  find  the  total 
area  of  the  geometrical  figures  by  mensuration,  measuring 
their  dimensions  not  by  means  of  the  cen- 
timetre scale  but  with  a  scale  copied  from 
the  ruling  of  the  notebook;  or  by  transfer- 
ring each  length  to  the  ruled  page  with  a 
pair  of  dividers,  so  as  to  obtain  the  area  in 
the  same  units  as  before. 

27.  Measurement  of  Volume. — Examine 
the  graduated  cylinder  and  the  graduated 
pipette  and  notice  the  volume  occupied  by 
one  cubic  centimetre  in  each.  Do  they 
seem  larger  or  smaller  than  the  space  that 
would  be  enclosed  if  an  imaginary  square 
centimetre  were  drawn  beside  one  linear 
centimetre  of  the  ruler  and  an  imaginary 
cube  were  then  built  up  on  the  square? 
Pour  into  a  test-tube  an  amount  of  water 
which  you  think  will  be  10  cm3;  then  meas- 
ure it  carefully.* 

Put  an  irregular  block  of  metal  (not  one  of 
the  standard  weights  from  your  set)  into  a 
graduated  cylinder  partly  filled  with  water 
and  determine  its  volume  by  the  change  in 
the  water  level.     In  reading  a  cylinder  or 
a  pipette  the  result  is  to  be  obtained  by 
noting  the  height  of  the  liquid  surface  where  it  is  lowest 
in  the  centre  (Fig.  4). 
28.  Measurement    of     Mass. — Examine     the    brass 

*  See  that  a  towel  is  at  hand  when  water  is  used  in  any  experi- 
ment, and  wipe  up  immediately  any  that  is  spilled  on  the  table. 


FIG.       4. 

G  R  A  DUATED 

CONTAINER. 
— The  scale  is 
customarily 
arranged  to 
give  the  cor- 
rect reading 
at  the  lowest 
part  of  the 
meniscus,  or 
curved  sur- 
face, of  the 
liquid. 


V 

II  WEIGHTS  AND  MEASURES  33 

weights  in  a  set  extending  from  one  gram  to  500  grams, 
and  observe  especially  the  size  of  the  10-gram  weight. 
What  would  its  volume  be  if  its  density  were  ten  (gm. 
per  cm3)?  If  the  density  of  brass  is  only  8.5  (i.  e.,  if 
it  is  less  compact)  will  its  volume  be  greater  or  less  than 
this? 

Examine  the  platform  balance  and  notice  that  there 
are  two  wooden  wedges  that  hold  the  pans  away  from 
the  beam  and  the  beam  away  from  its  support,  so  as  to 
remove  their  weight  from  the  accurately  ground  bearings 
when  the  apparatus  is  not  in  use.  With  a  fine  analytical 
balance  this  is  such  an  important  matter  that  a  mechan- 
ism is  provided  by  means  of  which  the  user  keeps  the 
scale  pans  and  the  beam  " supported"  while  arranging 
the  weights  and  the  object  to  be  weighed,  and  only  lowers 
them  upon  their  bearings  for  a  few  moments  at  the  time 
of  actual  weighing.  Such  care  is  not  necessary  with  an 
ordinary  platform  balance,  but  it  is  always  advisable  to 
support  the  scale  pans  after  one  has  finished  using  the 
apparatus.  Remove  the  wedges  carefully  and  notice 
where  the  moving  pointer  comes  to  rest  on  the  arbitrary 
scale.  This  need  not  be  in  the  centre  but  may  be  at 
any  point  on  the  scale,  and  in  weighing  if  the  weights  are 
so  added  as  to  return  the  pointer  to  this  same  position 
the  result  will  be  the  same  as  if  the  pointer  were  made  to 
take  a  central  position  in  the  first  place.  Notice  the 
counterpoise,  which  can  be  screwed  toward  one  side  or 
the  other  in  order  to  adjust  the  position  of  equilibrium, 
but  do  not  attempt  to  move  it  unless  you  are  sure  that 
the  apparatus  is  on  a  level  part  of  the  table  and  the  scale 
pans  are  free  from  dust  or  other  adherent  matter.  Notice 
whether  the  balance  has  a  sliding  weight  that  is  used  for 
weighing  fractions  of  a  gram. 

Find  the  mass,  in  grams,  of  a  four-ounce  avoirdupois 
4 


34  THEORY  OF  MEASUREMENTS  §30 

weight,  weighing  it  first  on  the  left  pan  of  the  balance  and 
then  on  the  right.  Notice  that  when  it  is  on  the  right- 
hand  pan  the  reading  of  the  sliding  weight  must  be 
subtracted  from  the  sum  of  the  other  weights  instead  of 
being  added  to  them. 

Draw  up  a  few  cubic  centimetres  of  water  in  the 
graduated  pipette,  retaining  it  by  pressing  the  dry  finger 
tip  on  the  upper  end  of  the  tube.  Practice  letting  it  run 
out  slowly  until  you  have  no  trouble  in  delivering  any 
exact  amount.  Then  fill  it  again,  weigh  the  ''catch- 
bucket"  or  some  other  container,  run  just  seven  cm3  of 
water  from  the  pipette  into  the  container  and  weigh  the 
latter  again. 

29.  Measurement    of    Density. — Calculate  the    den- 
sity of  water  from  the  data  obtained  in  the  last  experi- 
ment. 

Find  the  volume  of  the  200-gram  weight  by  measuring 
its  height  and  diameter  as  carefully  as  possible,  but  do 
not  immerse  it  in  water.  Suppose  that  the  handle  of  the 
weight  were  soft,  like  wax,  and  could  be  flattened  out  and 
spread  uniformly  over  the  top  of  the  cylindrical  part, 
and  estimate  as  well  as  you  can  how  much  this  would 
add  to  the  height  of  the  cylinder.  Then  calculate  the 
density  of  the  brass  weight,  remembering  that  multi- 
plication and  division  are  always  to  be  done  by  the 
abridged  methods.  The  result  may  turn  out  to  be  less 
than  the  usual  density  of  brass  (about  8.5  gm.  per  cm3) 
if  the  handle  is  a  separate  piece  screwed  into  the  body  of 
the  weight  so  that  an  air  space  is  left  between  the  two 
parts. 

Find  also  the  mass  of  the  irregular  solid  whose  volume 
was  determined  by  immersion  and  calculate  its  density. 

30.  Equivalents. — In  most  English-speaking  countries 
the  C.G.S.  System  is  very  little  used  except  for  scientific 


II  WEIGHTS  AND  MEASURES  35 

purposes,  the  older  system  still  holding  its  own  in  spite  of 
such  obvious  disadvantages  as  possessing  an  ounce 
(avoirdupois)  that  weighs  about  28  gm.  and  another 
ounce  (Troy)  that  is  equal  to  a  little  more  than  31  gm.; 
furthermore,  the  U.  S.  fluid  ounce  of  water  weighs  more 
than  an  avoirdupois  ounce  and  less  than  a  Troy  ounce, 
and  is  four  percent  larger  than  the  imperial  fluid  ounce 
of  England.  Accordingly,  a  knowledge  of  the  approxi- 
mate relationships  between  the  units  of  the  old  system 
and  the  new  is  almost  a  necessity  for  the  scientific  student 
of  today. 

Turn  to  the  tables  of  equivalent  weights  and  measures 
in  the  appendix  of  this  book  and  use  the  approximate 
equivalents  for  translating  your  own  weight  and  height 
into  the  C.G.S.  System,  and  for  answering  such  questions 
as:  How  many  kilometres  is  the  distance  from  here  to 
New  York  (or  any  other  city)  ?  What  is  the  approximate 
height  of  this  room  in  metres?  What  is  the  C.G.S. 
velocity  of  sound  if  it  travels  a  mile  in  five  seconds? 
How  many  centimetres  per  second  is  your  ordinary  rate 
of  walking? 

31.  Questions  and  Exercises. — 1.  Explain  why  round- 
ing off  a  half  to  the  nearest  even  number  will  increase  a 
measurement,  in  the  long  run,  just  as  often  as  it  will 
decrease  it. 

2.  When  you  drew  a  line  "just  ten"  centimetres  long 
was  its  length  10.0  cm.?     Was  it  10.00  cm.?     10.000  cm.? 
Measure  it  again,  and  explain  why  it  is  better  to  use  one 
of  these  numbers  in  stating  its  length  than  to  say  "just" 
ten. 

3.  Does  the  sliding  weight  on  the  platform  balance  add 
its  scale-indication  to  the  right  pan  or  to  the  left?     How 
can  it  give  correct  results  if  the  zero  is  not  at  the  centre 
of  its  scale? 


36  THEORY  OF  MEASUREMENTS  §31 

4.  What  disadvantage  is  there  if  the  finger-tip  that 
closes  the  top  of  the  measuring  pipette  is  not  dry? 

5.  The  specific  gravity  of  a  substance  is  denned  as  the 
ratio  of  its  mass  to  the  mass  of  an  equal  volume  of  water, 
i.  e.,  the  relative  density  compared  with  water  as  a  stand- 
ard.    How  do  the  specific  gravity  and  the  density  of  a 
substance  compare  if  the  density  of  water  is  1?     If  it  is 
a  little  less  than  one? 

6.  If  specific  volume,  v,  is  defined  for  any  substance  as 
the  volume  per  gram  of  mass,  what  will  the  equation  be 
that  shows  the  relationship  between  p  (density)  and  u? 

7.  If  10  cm.  =  4  inches  make  a  rapid  mental  calcula- 
tion of  the  C.G.S.  length  of  12  inches.     Of  40  inches;  of 
10  inches;  of  3  feet;  of  7  inches. 

8.  State  the  distance  of  10  kilometres  as  the  nearest 
whole  number  of  miles.     State  6  miles  as  the  nearest 
whole  number  of  kilometres. 

9.  Reduce  5  pounds  per  square  foot  (pressure)  to  gm. 
per    cm2.     Reduce   x   lbs./ft.2   to   gm./cm2.      Write    in 
your  notebook  directions  for  reducing  1  gros  (weight)  per 
square  pouce  (length)  to  grams  per  square  centimetre,  or 
for  reducing  prices  in  roubles  per  pood  to  cents  per 
kilogram,  and  submit  them  to  your  instructor  for  ap- 
proval. 


III.     ANGLES   AND   CIRCULAR  FUNCTIONS 

Apparatus. — A  pair  of  dividers;  a  pencil  compass;  a 
protractor;  a  ruler;  a  pencil  with  a  fine  point. 

32.  Unit  of  Angle. — When  two  straight   lines   inter- 
sect in  such  a  way  that  each  is  perpendicular  to  the  other 
the  amount  of  their  divergence,  is  said  to  be  ninety  degrees 
or  one  right  angle.     A  single  degree  is  then  a  compara- 
tively small  difference  indirection;  two  lines  drawn  from 
the  observer  toward  the  opposite  edges  of  the  sun's  disc 
include  an  angle  of  about  half  a  degree.     For  the  sake  of 
avoiding  fractions  at  a   time  when  decimals  were  not 
used,  the  degree  (1°)  was  divided  into  sixty  parts,  each 
called  one  minute  (I'),  and  a  sixtieth  of  a  minute  was 
used  as  a  still  smaller  unit,  one  second  (!")•     These  three 
units  are  still  in  use  all  over  the  world  for  expressing  the 
size  of  an  angle  as  a  " mixed"  denominate  number.     For 
example  the  circular  arc  which  is  of  the  same  length  as 
the  radius  corresponds  to  an  angle  of  57°  17'  45". 

33.  Circular  Measure. — If  the  original  right  angle  had 
been  divided  and  re-divided  into  tenths  or  hundredths 
instead  of  into  ninetieths   and  sixtieths  the  resultant 
units  would  have  been  much  more  convenient  for  pur- 
poses   of    calculation.     The    question    suggests    itself, 
however,  as  to  why  the  right  angle  should  be  arbitrarily 
chosen  for  the  quantity  that  is  to  be  subdivided.     Why 
not  take  the  whole  circumference  or  some  other  amount 
of  angle?     As  a  matter  of  fact,  this  arbitrariness  is  often 
avoided  for  scientific  purposes  by  measuring  an  angle  by 
an  entirely  different  method:     The  vertex  of  the  angle 
is  taken  as  a  centre,  around  which  an  arc  of  a  circle  is 
drawn  extending  from  one  side  of  the  angle  to  the  other 

37 


38  THEORY  OF  MEASUREMENTS  §34 

(Fig.  5).  The  length  of  this  arc  will  depend  not  only 
upon  the  size  of  the  angle  but  also  upon  the  length  of  the 
radius  that  is  used,  but  the  ratio  of  length  of  arc  to  length 


FIG.  5.  CIRCULAR  MEASURE. — The  ratio  of  any  arc  to  its  radius 
is  taken  as  the  numerical  or  circular  measure  of  the  angle  at  the 
centre.  DC/OC  =  BA/OA  =  3/4,  approximately,  for  the  angle 
represented  here. 

of  radius  will  depend  only  upon  the  size  of  the  angle  and 
so  can  be  used  as  a  measure  of  it.  In  the  diagram  the 
arc  AB  seems  to  be  about  f  as  long  as  the  radius  OA, 
the  arc  CD  is  likewise  f  as  long  as  the  radius  OC,  and  the 
circular  measure  of  this  particular  angle  is  accordingly 
f,  or  0.75. 

Draw  an  angle  which  is  somewhat  less  than  a  right 
angle.  Draw  its  arc,  and  measure  the  curved  line  as  well 
as  you  can  with  an  ordinary  ruler.  Measure  the  radius 
also,  and  calculate  the  circular  measure  of  the  angle.  It 
will  probably  turn  out  to  be  about  1.4.  Is  the  circular 
measure  of  a  right  angle  equal  to  just  1.5?  State  why. 

34.  Numerical  Measure  of  an  Angle. — Since  the 
size  of  an  angle  is  denned  as  the  quotient  arc  divided  by 
radius  it  follows  that  this  amount  is  not  a  number  of 
centimetres  or  of  any  other  arbitrary  units  but  is  a  pure 
number.  If  the  arc  measures  6  centimetres  and  the 


Ill  ANGLES  AND  CIRCULAR  FUNCTIONS  39 

radius  is  3  centimetres  the  size  of  the  angle  is  the  abstract 
number  2,  not  2  centimetres;  and  if  both  had  been 
measured  in  inches  the  quotient  would  still  have  been 
merely  the  number  2,  the  arc  being  two  times  the  radius, 
not  two  centimetres  times  the  radius.  The  expression 
numerical  measure  of  an  angle  has  the  same  meaning 
as  circular  measure  of  an  angle,  and  denotes  the  way  in 
which  the  size  of  an  angle  is  always  expressed  for  the- 
oretical purposes.  One  of  the  chief  advantages  of  this 
method  lies  in  the  simplification  which  it  causes.  Just 
as  the  foot-pound  system  of  measures  makes  the  density 
of  water  about  62  lb./ft3,  and  hence  makes  specific 
gravity  approximately  equal  to  density  divided  by  62, 
instead  of  merely  to  density  as  in  the  C.G.S.  System,  so 
angular  velocity  would  be  represented  by  57.28  v/r 
instead  of  v/r,  and  such  higher  mathematical  expressions 
as  Dx  sin  x  =  cos  x  and  Dx  cos  x  =  —  sin  x  would  become 
Dx  sin  x  =  .01746  cos  x  and  D£  cos  x  =  —  TT  sin  x/lSO 
if  the  angles  were  measured  in  degrees  instead  of  nu- 
merically. 

A  suggestion  of  the  reason  why  an  angle  ought  to  be 
expressed  as  an  abstract  number  instead  of  in  terms  of 
a  unit  may  be  obtained  by  imagining  a  length  of  one 
centimeter  and  an  angle  of  45  degrees  to  be  drawn  on  a 
sheet  of  paper  and  observed  through  a  magnifying  glass. 
The  centimetre  may  appear  to  be  enlarged  to  a  length 
of  two  centimetres,  but  the  angle  of  45  degrees  does  not 
become  an  angle  of  90  degrees;  it  remains  exactly  the 
same  size  as  before.  The  same  thing  is  of  course  true 
of  an  abstract  number:  with  a  very  slight  magnification 
two  objects  may  both  be  made  to  look  larger,  but  no 
amount  of  magnifying  power  will  make  them  look  like 
three. 

35.  The  Angle  TT  and  the  Unit  Angle. — Draw  an  angle 


40  THEORY  OF  MEASUREMENTS  §36 

of  180°  and  its  arc,  viz.,  a  semicircle.  Obviously  its 
numerical  measure,  semicircumference  divided  by  semi- 
diameter,  is  the  same  fraction  as  the  ratio  of  the  whole 
circumference  to  the  whole  diameter,  which  is  denoted 
by  the  symbol  TT  and  is  approximately  equal  to  3.1416. 
It  is  also  clear  that  an  angle  which  extends  entirely 
around  a  point,  that  is,  four  right  angles  or  360  degrees, 
must  have  2ir  for  its  numerical  measure;  one  right  angle 
must  be  equal  to  7r/2,  7r/4  is  the  same  as  45  degrees,  etc. 
If  180°  =  TT  the  numerical  measure  of  one  degree  must 
be  7T/180  and  the  degree-measure  of  an  angle  that  is 
numerically  equal  to  one  must  be  180/Tr. 

Find  the  number  of  degrees  in  the  unit  angle  by  writing 
the  value  of  TT,  carried  out  to  at  least  eight  or  ten  decimal 
places,  and  using  the  method  of  abridged  division  to 
find  out  how  many  times  it  is  contained  in  180.  The 
latter  number  should  be  written  180.00000.  .  .  with  as 
many  ciphers  as  may  be  necessary. 

Practice  translating  such  numbers  as  the  following  into 
degrees  until  it  can  be  done  without  any  hesitation: 

ITT    =    ?    27T    =    ?    JTT    =    ?    47T    =    ?    7T/3    =    ?    f  7T    =    ?    7T/4    =    ? 

7T/7T  =  ?  JTT  =  ?  If  an  angle  is  numerically  equal  to 
three  is  it  greater  or  less  than  180°? 

Practice  translating  the  following  numbers  of  degrees 
into  circular  measure  until  it  can  be  done  fluently: 
90°  =  ?  360°  =  ?  45°  =  ?  3  right  angles  =  ?  180°  =  ? 
30°  =  ?  60°  =  ?  1°  =  ?  270°  =  ?  57°.296  =  ? 

36.  The  Protractor. — A  protractor  is  a  scale  of  angles 
just  as  a  graduated  ruler  is  a  scale  of  lengths.  It  consists 
essentially  of  a  zero  line  on  which  is  a  point  that  rep- 
resents the  vertex  of  the  angle,  and  a  curved  scale  of 
short  lines  so  placed  that  if  they  were  sufficiently  long 
each  one  would  pass  through  the  vertex  and  make  an 
angle  with  the  base  line  equal  to  the  number  of  degrees 
with  which  it  is  marked. 


Ill 


ANGLES  AND  CIRCULAR  FUNCTIONS 


41 


Examine  the  protractor  and  its  scale  of  degrees. 
Notice  that  the  centre  toward  which  the  slanting  lines 
converge  must  be  on  the  line  joining  0°  and  180°  and 
hence  must  be  at  the  top  of  the  notch  shown  in  the  figure. 
It  is  always  at  the  corner  that  is  made  rectangular,  the 


FIG.  6.  PROTRACTOR. — The  essential  parts  of  a  protractor  are 
the  zero  line,  the  central  point,  and  the  scale  showing  where  a  line 
must  pass  to  form  the  second  side  of  any  required  angle. 

other  being  obtuse  or  rounded.  To  draw  a  line  which 
shall  make  a  given  angle  with  a  given  line  at  any  particu- 
lar point  the  protractor  is  placed  so  that  its  zero  coincides 
with  the  given  line  and  its  centre  with  the  given  point. 
A  fine  line  or  dot  is  then  made  opposite  the  required 
number  of  degrees  on  the  scale,  the  protractor  is  re- 
moved, and  a  ruler  is  used  to  draw  a  straight  line  through 
the  dot  and  the  particular  point  on  the  given  line. 

Use  the  protractor  to  draw  a  triangle  of  any  convenient 
size  and  of  such  shape  that  its  three  angles  are  7r/2,  7r/3, 
and  7T/6.  Draw  another  so  that  its  angles  are  ir/4,  ir/4, 
and  7T/2. 

37.  The  Diagonal  Scale. — If  the  protractor   is   pro- 


42  THEORY  OF  MEASUREMENTS  §38 

vided  with  what  is  known  as  a  diagonal  scale  notice  that 
at  the  top  of  this  scale  there  is  a  horizontal  line 
which  is  divided  into  centimetres  or  inches  and  that  one 
division  at  the  end  of  the  scale  is  divided  into  tenths. 
Decide  for  yourself  how  it  is  that  any  length,  such  as  7.4, 
within  the  limits  of  the  total  length  of  the  scale,  can  be 
found  already  laid  off  as  a  single  continuous  stretch  of 
the  base  line,  the  tenths  being  measured  from  the  proper 
point  to  the  junction  of  the  tenths'  scale  and  the  units' 
scale,  and  the  units  then  extending  onward  the  required 
distance  beyond  the  junction  point.  Notice  that  the 
tenths'  divisions  are  prolonged  downward  so  as  to  cut  diag- 
onally across  parallel  horizontal  lines  and  shift  a  single 
tenth  to  one  side  while  dropping  ten  lines  downward. 
This  means  that  on  the  first  level  below  the  base  line 
their  shift  will  be  only  one  hundredth,  and  on  successive 
levels  will  be  .02,  .03,  ...  .10,  as  the  student  can  easily 
prove  for  himself  by  means  of  the  principles  of  similar 
triangles.  Accordingly,  any  number  of  units,  tenths, 
and  hundredths  can  be  found  marked  off  along  the  proper 
level.  Thus  a  distance  of  7.43  will  be  found  on  the 
third  level  between  the  same  diagonal  and  vertical  line 
as  mark  off  7.4  on  the  base  line.  A  pair  of  dividers  should 
be  used  to  span  an  unknown  length  and  then  transfer 
it  to  the  diagonal  scale  for  measurement,  or  to  take  a 
required  length  from  the  scale  for  the  purpose  of  laying 
it  off  on  paper.  They  should  be  held  rather  flat  against 
the  scale,  not  perpendicularly,  so  as  to  avoid  marring  it. 
38.  Measures  of  Inclination. — If  a  slanting  line  inter- 
sects a  level  line  the  inclination  of  the  former  may  be 
measured  either  by  the  size  of  the  angle  between  them 
or  by  the  rise  of  the  inclined  line  per  unit  of  level  distance 
(see  §  17).  The  second  of  these  two  amounts  is  said  to 
be  the  tangent  of  the  first  one;  thus  2/5  =  tan  22°.  This 


Ill 


ANGLES  AND  CIRCULAR  FUNCTIONS 


43 


relationship  can  easily  be  generalized  so  as  to  include 
cases  in  which  neither  side  of  the  angle  is  horizontal: 
From  any  point  on  one  side  of  an  angle  draw  a  straight 
line  perpendicular  to  the  other  side;  this  will  complete  a 
right-angled  triangle,  and  can  be  done  in  all  cases  where 
the  angle  is  acute  (i.  e.,  between  0°  and  90°  in  size).  The 
tangent  of  an  acute  angle  of  any  right-angled  triangle  is 
defined  as  the  ratio  of  the  side  opposite  the  angle  to  the 
side  that  is  adjacent  to  the  angle,  the  word  "side"  being 
used  here  to  indicate  either  of  the  right-angled  sides,  not 
the  hypothenuse. 

In  the  diagram  (Fig.  7)  the  angle  G  is  an  acute  angle  of 


FIG.  7.  ANGLES  AND  THEIR  TANGENTS. — The  ratio  of  any  vertical 
side  to  the  corresponding  base  is  a  number  which  is  called  the 
tangent  of  the  acute  angle  at  the  base.  EF  is  the  same  fraction  of 
GF  as  AB  is  of  GB,  namely  about  1/3.  PQ  is  about  twice  OQ, 
and  RS  is  about  twice  OS.  Accordingly  tan  G  =  1  ,/3  and  tan  O  =  2. 

each  of  three  different  right-angled  triangles.  Measure 
the  height  and  the  base  of  any  one  of  these  and  calculate 
the  tangent  of  the  angle  G.  The  reason  for  the  name 
"tangent"  may  be  understood  by  noticing  that  the  line 
drawn  between  the  two  sides  of  the  angle  and  tangent  to 
one  end  of  the  arc  will  be  numerically  equal  to  the  ratio 
in  question  if  the  radius  GD  or  GE  is  of  unit  length,  for 
then  CD/GD  =  CD/I  =  CD. 


44  THEORY  OF  MEASUREMENTS  §39 

The  inclination  of  two  lines  that  form  an  acute  angle 
of  a  right-angled  triangle  may  also  be  measured  by  the 
ratio  of  the  opposite  side  to  the  hypothenuse.  This  is 
called  the  sine  of  the  angle;  for  example,  in  Fig.  7,  the 
ratio  of  EF  to  GE,  or,  what  amounts  to  the  same  thing, 
EF/GD,  is  the  sine  of  the  angle  G,  often  abbreviated  to 
"sin  G."  It  is  evident  that  these  definitions  of  sine 
and  tangent  agree  with  those  that  have  been  previously 
given  for  the  " gradient"  and  ''per  cent  slope"  of  an 
angle  between  a  sloping  line  and  a  horizontal  one. 

39.  Use  of  a  Table  of  Tangents. — Close  to  the  bottom 
of  the  next  unused  page  of  your  notebook  draw  a  fine 
horizontal  line  having  a  length  of  either  20,  25,  or  50  of 
the  squares  made  by  the  cross-lines  of  the  paper.  Draw 
it  directly  on  one  of  the  horizontally  ruled  lines,  pref- 
erably so  that  it  extends  nearly  across  the  page.  Call 
its  length  unity  (1);  it  may  not  be  one  footer  one  deci- 
metre, but  it  is  to  be  considered  as  having  a  length  of 
just  one  arbitrary  unit,  which  need  not  be  given  any 
name.  Mark  a  figure  1  under  its  right-hand  end,  a 
figure  0  at  the  left-hand  end,  and  the  scale  of  numbers 
0.1,  0.2,  0.3,  etc.,  at  intervals  of  two,  or  two  and  a  half, 
or  five,  squares,  as  may  be  required  by  the  length  of  the 
line.  From  the  right-hand  end  of  this  base  line  draw  a 
fine  line  perpendicular  to  it,  extending  it  as  far  as  the 
top  of  the  page.  Beginning  with  zero  at  the  junction 
with  the  base  line  lay  off  a  similar  scale  along  the  vertical 
line.  Do  not  number  the  successive  squares  of  the  paper 
1,  2,  3,  etc.,  but  see  that  this  scale  indicates  the  same 
proportions  as  are  given  by  the  horizontal  one.  Turn  to 
the  table  of  circular  functions  in  the  appendix;  look  for 
the  number  10  in  the  column  headed  DEG  and  notice 
that  the  number  opposite  it  in  the  column  TAN  is  1763; 
this  means  that  the  tangent  of  10°  is  .1763.  On  the 


Ill  ANGLES  AND  CIRCULAR  FUNCTIONS  45 

vertical  line  just  drawn  make  a  small  mark  at  a  height 
of  .1763  above  the  base  line  according  to  the  vertical 
scale  already  made.  In  the  same  way  lay  off  tan  20° 
on  the  same  scale.  The  decimal  points  are  omitted  from 
the  table;  but  the  tangent  of  a  large  angle  is  obviously 
greater  than  that  of  a  smaller  angle,  and  the  table  shows 
that  for  successive  degrees  the  tangent  increases  gradu- 
ally and  quite  regularly.  Notice  that  the  tangent  of  45° 
is  1.000000.  .  .,  and  that  angles  and  tangents  larger  than 
this  must  be  sought  for  above  the  abbreviations  DEG  and 
TAN  which  are  at  the  bottom  of  the  page.  If  there  is 
any  trouble  in  understanding  how  the  table  is  arranged 
use  it  to  verify  the  following  equations  before  proceeding 
further : 

Tan  1°  =  .0175;  tan  2°  =  .0349;  tan  6°  =  .1051; 
tan  44°  =  .9657;  tan  45°  =  1.000;  tan  46°  =  1.036; 
tan  84°  =  9.514;  tan  85°  =  11.43;  tan  89°  =  57.29. 

Lay  off  tan  30°,  tan  40°,  tan  50°,  etc.,  as  far  as  the 
length  of  the  vertical  line  will  allow;  then  draw  slanting 
lines  from  each  of  these  points  to  the  left-hand  end  of 
the  base  line.  With  the  protractor  test  the  angles  formed 
in  order  to  make  sure  that  they  are  accurately  10°,  20°, 
30°,  etc.  If  mistakes  have  been  made  repeat  the  con- 
struction on  the  next  page;  do  not  correct  the  first 
diagram  by  erasures. 

40.  Experimental  Determination  of  Sines. — With  the 
base  line  as  a  radius  and  its  left-hand  end  as  a  centre  draw 
an  arc  on  the  diagram  that  has  just  been  made,  extending 
it  from  0°  to  90°.  Complete  the  series  of  angles  as  far 
as  90°  by  laying  off  successive  ten-degree  arcs  or  chords 
with  a  pair  of  dividers.  Find  the  point  where  the  line 
whose  slope  is  10°  intersects  the  arc,  and  note  carefully 
the  vertical  distance  from  the  base  line  up  to  this  point, 
but  do  not  draw  a  vertical  line.  If  the  diagram  has  been 


46 


THEORY  OF  MEASUREMENTS 


§41 


angle 

sine 

0° 
10° 
20° 
30° 
40° 

0 
.175  -  .001 
.345  -  .003 
.500  +  .000 
.640  +  .003 

carefully  drawn  with  a  sharp-pointed  pencil  the  value 
should  come  out  0.174,  being  measured  of  course  in  terms 
of  the  figured  scales,  not  in  terms  of  the  small  ruled 
squares.  Notice  that  this  number  is  less  than  tan  10°, 
and  that  it  corresponds  to  the  ratio  EF/GD  in  Fig.  7, 
and  hence  is  the  sine  of  the  angle  10°.  In  the  same  way 
measure  sin  20°,  sin  30°,  ...  sin  90°,  or  as  many  of  them 
as  may  be  directed  by  the  instructor,  and  tabulate  the 
results  in  the  first  two  columns  of  a  three-column  table. 

Then  turn  to  the  table  of  cir- 
cular functions,  find  the  true 
numerical  values  of  the  sines 
from  the  column  headed  SIN,  and 
correct  your  measured  sines  by 
adding  a  third  column  as  shown 
here;  the  sine  of  40°,  for  ex- 
ample, appears  to  have  been 
measured  as  .640  and  then 
found  to  be  actually  .643, 
whence  the  correction  of  +  .003 
in  the  third  column.  After 
your  table  of  sines  has  been 

completed  find  the  angle  whose  sine  has  the  largest  cor- 
rection and  divide  this  correction  mentally  by  the  true 
value  of  the  sine  in  order  to  find  the  relative  error  of  the 
measurement.  Thus,  if  the  quotient  is  about  3/600 
=  1/200  =  .005  your  measurement  has  an  error  that 
amounts  to  three  parts  out  of  a  total  of  600,  which  is  the 
same  as  5  per  thousand,  or  f  percent.  Call  your  error 
" moderate"  if  it  is  anywhere  between  0.3  percent  and 
1  percent;  call  it  " large"  if  greater  than  1  percent,  and 
" small"  if  less  than  0.3  percent. 

41.  Definition  of  Function. — Up  to  the  present  point  it 
has  been  assumed  that  an  angle  of  any  size  from  0  to 


TABLE  OF  SINES. — The 
second  column  shows  the 
measured  value;  the  third 
column  is  the  amount  of 
change  that  must  be  made 
to  obtain  the  true  value. 


Ill  ANGLES  AND  CIRCULAR  FUNCTIONS  47 

7T/2  or  90°  will  have  a  tangent  which  is  a  definite  number 
for  each  definite  size  of  angle.  A  quantity  which  can  be 
assumed  to  have  different  sizes,  whether  restricted  to  a 
certain  range  or  not,  is  called  a  variable;  and  a  second 
quantity,  which  in  general  has  a  definite  value  for  each 
particular  value  of  the  first,  is  said  to  be  a  function  of  that 
variable.  For  example  x2  —  3x  +  2  is  called  a  function 
of  x,  because  it  has  a  definite  value  for  any  definite  value 
that  may  be  assigned  to  x.  Similarly  sin  x,  tan  x,  V  #, 
logarithm  of  x,  xn,  ax,  are  all  functions  of  x,  as  is  also 
any  other  algebraical  formula  which  involves  x. 

42.  The  Cosine  of  an  Angle. — Another  function   of 
an  angle  which  is  frequently  useful  is  the  cosine.     In  a 
right  triangle,  such  as  was  used  in  defining  the  tangent 
and  the  sine  of  an  angle,  it  is  the  ratio  of  the  adjacent 
side  to  the  hypothenuse;  the  ratio  of  GF  to  GE,  in  Fig.  7, 
is  the  cosine  of  the  angle  G,  or,  as  it  is  usually  abbreviated, 
cos  G  =  GF/GE. 

43.  Circular  Functions. — The  sine,  cosine,  and  tangent 
are  included  under  the  general  term  circular  functions,  a 
phrase  which  includes  also  three  other  functions  of  an 
angle  which  are  not  so  frequently  used.     These  are  the 
cotangent,  the  secant,  and  the  cosecant,  and  may  be  de- 
fined by  the  equations  cot  x  =  I/tan  x,  sec  x  =  I/cos  x, 
and  cosec  x  =  I/sin  x.     Another  set  of  definitions,  which 
are  perhaps  more  interesting  and  easier  to  remember,  may 
be  obtained  by  the  use  of  a  circle  diagram  and  extended 
so  as  to  include  angles  greater  than  90° :  For  the  sake  of 
uniformity  the  angle  is  so  placed  that  one  of  its  sides 
extends  out  horizontally  to  the  right.     A  zero   angle 
would  have  its  second  side  coincident  with  the  first,  and 
larger  angles  may  be  supposed  to  have  been  generated 
by  rotating  the  second  side  through  the  required  angular 
distance  from  the  first.     The  usual  convention  is  that 


48 


THEORY  OF  MEASUREMENTS 


§43 


the  direction  of  rotation  shall  be  counter-clockwise,  i.  e., 
in  the  opposite  direction  to  that  in  which  the  hands  of  a 
clock  turn.  A  circle  whose  radius  is  unity  is  drawn  around 
the  vertex  of  the  angle,  as  in  Fig.  8,  which  shows  an  angle 

of  somewhat  less  than 
45°.  A  line  (DB)  is 
drawn  upward  from  the 
right-hand  end  (D)  of 
the  horizontal  radius, 
and  a  perpendicular 
(AC)  is  dropped  from 
the  end  of  the  inclined 
radius  (A)  to  the  base 
line  (OD).  Then  the 
length  of  the  tangent 
to  the  circle  (DB)  is 
the  numerical  tangent 
of  the  angle  (0),  the 
line  (OB)  that  cuts 
the  circle  is  the  secant 


FIG.  8.  CIRCULAR  FUNCTIONS. — 
DB  =  tan  DO  A;  AC  =  sin  DO  A; 
OC  =  cos  DO  A;  OB  =  sec  DO  A. 
The  radius  is  supposed  to  be  of 
unit  length. 


(Lat.  secans,  cutting), 
and  the  perpendicular 
(AC),  which  cuts  off  a  rounded  hollow  of  the  figure,  is 
called  the  sine  (Lat.  sinus,  a  bay).  The  cosine  is  the 
sine  of  the  complementary  angle  (e.  g.,  Z  OAC',  if  x  is 
acute  cosx  =  sin  (90°  —  x)),  and  the  cotangent  and  cose- 
cant are  similarly  the  tangent  and  secant  respectively 
of  the  complementary  angle,  two  angles  being  called 
complementary  when  their  sum  is  7r/2  or  90°.  As  OC 
is  considered  to  be  a  positive  amount  when  measured  to 
the  right  of  0  it  is  only  natural  to  consider  the  cosine  as  a 
negative  number  when  C  is  to  the  left  of  0.  Likewise 
CA  or  DB  must  be  negative  if  below  the  base  line  instead 
of  above.  AO  and  DO  are  to  be  produced  if  necessary. 


Ill 


ANGLES  AND  CIRCULAR  FUNCTIONS 


49 


If  tan  45°  =  +  I  and  sin  45°  =  +  .707  what  are  the 
values  of  tan  (90°  +  45°)  and  sin  (90°  +  45°)?  Ans.: 
Tan  135°  =  -  1;  sin 
135°  =  +  1. 

What  is  the  cosine  of 
135°?  What  are  the 
tangent,  sine,  and  cosine 
of  180°  +  45°?  Of  270° 
+  45°? 

44.  Generalized  Idea 
of  Angle.  —  For  some 
purposes  the  term  angle 
may  be  defined  so  as  to 
include  only  angles  less 
than  7T/2.  For  other 
purposes  obtuse  angles 
(i.  e.,  up  to  TT  or  180°) 
may  need  to  be  in- 
cluded. The  circular 
functions  have  been  "al- 
ready defined  for  angles 


FIG.  9.  FUNCTIONS  OF  ANY 
ANGLES. — The  cosine  is  negative  if 
C  is  to  the  left  of  0;  the  sine  and 
tangent  are  negative  if  A  and  B 
are  below  the  level  of  0;  DO  and 
AO  are  produced  if  necessary. 


of  all  sizes  up  to  2ir 
(6.28,  or  360°),  but  it  is 
obviously  unnecessary  to 
stop  at  this  figure.  By 
supposing  the  line  OA 
in  Fig.  8  to  have  made  more  than  a  complete  revolution 
it  can  be  seen  that  the  sine,  cosine,  tangent,  etc.,  of 
360°  +  40°  must  all  have  the  same  values  as  the  corre- 
sponding functions  of  40°.  In  general,  any  circular 
function  of  any  angle  x  is  equal  to  the  same  function  of 
2ir  +  x,  or  of  4ir  +  x,  or  of  2mr  +  x  if  n  is  any  whole 
number.  A  negative  angle  is  of  course  one  that  is  gener- 
ated by  a  clockwise  rotation  from  the  position  of  the 
5 


50  THEORY  OF  MEASUREMENTS  §45 

base  line;  thus  the  angle  shown  in  Fig.  9  may  be  con- 
sidered either  as  +  240°  or  as  —  120°,  and  obviously  the 
circular  functions  of  any  negative  angle  —  x  have  the  same 
values  as  those  of  the  positive  angle  2ir  —  x.  Negative 
angles  have  to  be  considered  occasionally,  just  as  negative 
heights  or  lengths  need  to  be.  Angles  larger  than  +  2?r 
commonly  come  under  consideration  in  connection  with 
rotatory  motion.  Such  objects  as  a  spinning  top,  a 
fly-wheel,  a  planet,  do  not  commonly  move  through  an 
angle  less  than  360°  and  then  stop,  but  their  angular 
motion  may  be  of  almost  any  amount  according  to  the 
extent  of  time  occupied  by  it. 

45.  Questions  and  Exercises. — 1.  How  can  it  be 
proved  that  the  ratio  which  gives  the  numerical  measure 
of  an  angle  will  have  the  same  value  whether  the  radius 
is  long  or  short? 

2.  Translate   135°  into  circular  measure.     Make  an 
approximate  mental  calculation  of  the  number  of  degrees 
in  an  angle  whose  numerical  measure  is  6. 

3.  After  drawing  a  triangle  whose  angles  were  meant 
to  be  7T/2,  7T/3,  and  7r/6  what  would  you  do  if  you  found 
one  of  its  angles  inaccurate  but  the  other  two  correct? 

4.  The  tangent  of  80°  is  given  in  the  table  as  5671. 
Where  do  you  place  its  decimal  point  and  why? 

5.  The  steepness  of  a  slope  is  the  characteristic  which 
corresponds  to  its  sine;  a  larger  sine  means  a  steeper  slope. 
What  characteristic  can  you  name  that  will  correspond 
to  its  cosine? 

6.  Any  radius  of  a  rotating  wheel  describes  an  angle 
which  increases  steadily  from  0  to  (say)  407r.     Explain 
how  the  sine  of  this  variable  angle  behaves  during  the 
same  time. 

7.  What  is  the  approximate  value  of  tan  (—  91°)?     Of 
tan  (+89°)?     Of  tan  (-  89°)? 


Ill  ANGLES  AND  CIRCULAR  FUNCTIONS  51 

8.  What  is  the  approximate  value  of  the  tangent  of  the 
angle  A  in  Fig.  7?     How  does  it  compare  with  the  value 
of  the  tangent  of  the  angle  (r?     What  relationship  is 
there  between  tan  T  and  tan  0? 

9.  The  cotangent  of  any  angle,  A,  has  been  denned  as 
the  reciprocal  of  tan  A,  and  also  as  the  value  of  tan 
(90°  —  A).     Prove  that  these  definitions  are  identical 
if  A  <  90°  by  drawing  a  right  triangle  having  A  for  one 
of  its  acute  angles. 

10.  Draw  a  right  triangle  and  prove  that,  if  A  <  90°, 
sin2  A  +  cos2  A  =  1  using  the  theorem  of  Pythagoras  that 
the  square  of  the  hypothenuse  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides.* 

11.  Make  A  an  acute  angle  of  a  right  triangle  and 
prove  that  tan  A  =  sin  A/cos  A. 

12.  Using  a  circle  diagram  prove  that  in  general  for  an 
angle  of  any  size,  x, 

(a)  tan  x  =  sin  x/cos  x 

(6)  sin2  x  +  cos2  x  =  I 

(c)  tan  (90°  -  x)  =  I/tan  x 

(d)  cos  x  =  sin  (90°  -  x)  =  sin  (90°  +  x) 

13.  What  synonyms  have  you  already  learned  for  the 
sine  and  the  tangent  of  an  angle? 

*  Sin2  A  is  the  conventional  abbreviation  of  (sin  A)2,  not  of 
sin  (sin  A);  similarly  sin3  A,  etc.  Sin"1  A,  however,  is  always  used 
to  denote  the  angle  whose  sine  is  A,  not  I/sin  A. 


IV.     SIGNIFICANT  FIGURES 

Apparatus.— Scale  of  centimetres  and  millimetres; 
card  or  strip  of  paper;  circular  brass  measuring  disc. 

46.  Estimation  of  Tenths. — It  sometimes  happens 
that  a  measurement  requires  but  a  slight  degree  of 
accuracy,  and  time  and  trouble  can  be  saved  by  making 
it  only  roughly.  As  a  general  principle,  however,  it  is 
advantageous  to  make  all  measurements  as  accurately 
as  possible.  Thus,  measurements  of  length  made  with 
the  metre  stick  should  be  expressed  not  merely  to  the 
nearest  centimetre  but  to  the  nearest  millimetre  or  tenth 
of  a  centimetre.  This  is  not  the  best  that  can  be  done, 
however.  After  noticing  how  each  centimetre  of  the 
scale  is  divided  in  half  by  a  long  line  and  each  half  is 
subdivided  into  fifths  by  four  short  lines,  thus  indicating 
tenths  of  a  centimetre,  it  is  not  difficult  to  imagine  each 
of  the  millimetre  intervals  of  the  scale  divided  in  the 
same  way  into  tenths  of  a  millimetre  and  to  make  a  fairly 
good  estimate  of  just  how  many  of  these  parts  are  in- 
cluded in  the  length  that  is  to  be  measured.  Experienced 
observers  even  attempt  to  make  a  mental  subdivision 
into  hundredths  of  the  smallest  intervals  on  a  graduated 
scale,  and  find  that  it  is  only  occasionally  that  one  man's 
estimate  will  differ  from  another's  by  more  than  one  or 
two  hundredths,  but  for  the  beginner  even  the  estimation 
of  tenths  will  be  a  rather  uncertain  process.  To  gain 
proficiency  it  will  be  found  better  to  begin  with  larger 
subdivisions,  such  as  a  scale  of  centimetres  that  has  no 
millimetre  marks.  The  position  of  a  mark  placed  at 
random  on  such  a  scale  can  be  estimated  mentally  and 
the  accuracy  of  such  a  determination  can  then  be  tested 

52 


IV  SIGNIFICANT  FIGURES  ^  53 

by  actual  measurement  with  a  more  finely  divided  scale. 
47.  Practice  in  Estimating  Tenths. — Draw  a  short 
line  at  right  angles  to  the  edge  of  a  card  or  slip  of  paper 
(Fig.  10)  and  hold  this  edge  on  a  scale  of  centimetres  and 
millimetres  in  such  a  way 
that  the  smallest  gradua- 
tions are  hidden  but  the 
marks  indicating  centime- 
tres and  half-centimetres  FIG.  10.  ESTIMATING  TENTHS 
are  visible  Notice  the  OF  A  CENTIMETRE.— The  card  is 
half-centimetre  space  in  laid  on  the  scale  at  random,  but 

in  such  a  wav  as  to   hide  the 
which  the  cross  line  comes     gmall  dividing  lines     The  loca_ 

and  mentally  divide  it  into  tion  of  the  arrow,  in  millimetres, 
five  equal  parts  by  four  im-  is  then  guessed,  and  afterward 
aginary  lines  so  as  to  make  verified  by  sliding  the  card  down- 

an  estimate  of  the  location     ward  enoueh  to  exP°se  the  whole 

,,  ,        scale, 

of   the   line    on    the   card. 

For  example,  in  the  figure  the  arrow  seems  to  be  either 
three  fifths  or  four  fifths  of  the  way  from  the  scale- 
division  19.5  to  the  division  20.0,  making  its  position 
19.8  or  19.9,  but  it  may  be  difficult  to  decide  which  of 
these  numbers  is  the  nearer  without  actual  measurement. 
In  practice,  however,  the  estimate  should  be  written 
down  and  the  card  should  then  be  allowed  to  slide  care- 
fully across  the  ruler  until  the  millimetre  scale  is  just 
exposed.  The  position  on  the  scale  which  the  line  on  the 
card  occupies  will  then  be  ascertained  and  should  be 
written  in  the  notebook  beside  the  previous  estimate. 
The  statement  should  be  correct  to  the  nearest  milli- 
metre, without  any  effort  to  decide  upon  fractions  of  a 
millimetre  (see  §§23  and  24). 

Make  ten  such  estimates  with  the  card  placed  any- 
where along  the  scale  at  random,  and  tabulate  the  de- 
terminations-and  the  verifications  in  two  parallel  columns. 


54  THEORY  OF  MEASUREMENTS  §48 

Then  hold  the  card  a  trifle  higher,  so  as  to  hide  the  half- 
centimetre  graduations  as  well  as  the  millimetres  and 
make  twenty  more  determinations.  These  will  also  be 
estimates  of  the  nearest  millimetre,  but  will  require  the 
more  difficult  process  of  deciding  upon  tenths  of  a  whole 
centimetre  instead  of  fifths  of  a  half-centimetre. 

If  the  scale  that  is  used  in  the  foregoing  exercise  is  on 
the  lower  edge  of  a  metre  stick  there  will  probably  be  a 
duplicate  scale  along  its  upper  edge,  with  the  help  of 
which  it  would  be  easy  to  make  each  estimate  absolutely 
correct  before  verifying  it.  This  furnishes  a  good  illus- 
tration of  the  statement  that  the  student  should  aim  to 
learn  rather  than  to  do  (2).  Finding  out  the  correct 
position  of  a  line  over  a  hidden  scale  is  something  that  is 
utterly  useless  to  him  or  to  anyone  else ;  it  is  the  learning 
to  divide  a  centimetre  into  imagined  tenths  that  will  be 
valuable  to  him  later  when  it  becomes  necessary  for  him 
to  divide  a  millimetre  into  tenths  without  aid  and  without 
the  possibility  of  later  verification.  In  making  any  kind 
of  a  measurement  care  should  be  taken  to  avoid  any  extra- 
neous influences,  or  bias  or  prejudice  of  any  kind.  In 
the  present  case,  the  upper  scale  on  the  metre  stick 
should  be  kept  covered  in  some  way  unless  the  student  is 
sure  of  his  ability  to  disregard  it  completely  while  making 
his  estimations. 

48.  Mistakes  in  Estimating  Tenths. — Omitting  the 
preliminary  estimate  of  fifths,  examine  your  table  of 
estimated  tenths  closely  and  find  out  what  kind  of  error 
you  are  most  apt  to  make.  Some  students  find  it  hardest 
to  estimate  0.3  and  0.7  correctly;  others  have  almost  a 
uniform  tendency  to  read  a  position  like  12.0  as  either 
11.9  or  12.1.  The  latter  mistake  is  due  to  the  fact  that  a 
minute  deviation  from  the  position  of  a  visible  graduation 
is  very  easily  noticed  and  there  is  a  tendency  to  consider 


IV  SIGNIFICANT  FIGURES  55 

it  as  a  single  tenth.  Of  course  if  it  amounts  to  more  than 
half  of  a  tenth  this  is  correct;  but  if  it  is  less  than  half  a 
tenth  it  should  be  considered  as  0.0  instead  of  0.1.  The 
same  bias  may  even  cause  a  tendency  to  read  0.1  and 
0.9  as  0.2  and  0.8.  On  the  other  hand  there  may  be  just 
the  opposite  error  if  the  graduated  lines  are  rough  or 
coarse,  unless  the  observer  is  careful  to  estimate  from  the 
imaginary  centre  of  such  a  line  instead  of  from  its  margin. 

If  a  definite  kind  of  error  is  evident  from  a  study  of 
your  table  see  if  it  can  be  overcome  when  making  another 
short  series  of  determinations.  -Then  draw  a  second  line 
on  the  card,  place  the  latter  in  position  as  before,  estimate 
both  points,  and  find  the  distance  between  them  by  sub- 
traction. This  is  the  customary  method  of  measuring  a 
length,  and  is  preferable  to  making  one  line  coincide 
exactly  with  a  scale  division  and  estimating  only  the 
other  one,  in  spite  of  an  obvious  additional  source  of 
error. 

49.  Value  of  TT.— An  experimental  determination  of  the 
value  of  the  constant,  TT,  can  be  made  by  rolling  the  brass 
disc  along  a  metre  stick  to  find  the  length  of  its  circum- 
ference, then  measuring  its  diameter  and  calculating 
the  ratio.  Hold  the  disc  loosely  at  its  centre,  using  the 
thumb  and  forefinger  only.  Start  it  with  its  marked 
radius  on  some  definite  graduation  of  the  scale  and  roll  it 
in  a  straight  line  until  the  radius  again  comes  vertically 
down  on  the  scale.  Read  this  second  position,  remember- 
ing not  to  be  satisfied  with  the  nearest  millimetre  (0.1 
cm.)  but  to  make  as  good  an  estimate  as  possible  of  tenths 
of  a  millimetre  in  order  that  the  circumference  may  be 
correctly  measured  to  a  hundredth  of  a  centimetre.  Use 
the  metre  stick  to  measure  the  diameter  of  the  disc  with 
the  greatest  possible  care,  avoiding  the  end  of  the  stick, 
which  may  be  a  little  worn  so  that  it  does  not  represent 


56  THEORY  OF  MEASUREMENTS  §51 

precisely  0.00  cm.  or  100.00  cm.  Find  the  value  of  IT 
by  dividing  circumference  by  diameter,  and  in  this  partic- 
ular case  using  the  unabridged  method  of  division  and 
carrying  out  the  result  until  it  has  two  or  three  figures 
that  are  different  from  the  theoretical  value  of  IT,  which 
is  3.141592653589793238462643383279502884197169399. 
Finally  round  off  (§§  23;  24)  both  your  result  and  the  true 
value  to  the  same  number  of  places,  choosing  that 
number  so  that  your  result  will  show  just  one  incorrect 
figure;  for  example,  3|  (or  3.1428571)  will  have  just  one 
wrong  figure  if  it  is  rounded  off  to  3.143,  because  the 
theoretical  value  will  round  off  to  3.142;  while  3.14234567 
should  be  rounded  off  to  3.1423  to  compare  with  the 
correct  value  3.1416. 

50.  Physical  Measurement. — The  operation  of  mak- 
ing a  measurement  is  merely  counting;  it  is  the  deter- 
mination of  how  many  units  of  a  certain  kind  are  required 
in  order  to  be  equal  to  a  given  quantity  of  the  same  kind. 
But  while  a  count  such  as  a  census  of  the  number  of 
individuals  in  a  town  must  give  a  perfectly  definite  whole 
number  it  usually  happens  that  a  physical  measurement 
will  not  give  a  whole  number,  or  even  a  commensurable 
number  except  as  the  result  of  an  error,  and  successive 
repetitions  of  a  measurement  will  give  a  number  of  dif- 
ferent apparent  values.     (Try  it;  measure  the  circum- 
ference of  the  7r-disc  a  second  time.)     Accordingly,  any 
numerical  statement  of  a  measurement  must  be  merely 
an  approximation  to  an  unknown  true  value,  and  so 
will  be  either  indistinguishably  correct  or  perceptibly 
incorrect  accoiding  to  how  closely  it  can  be  examined. 

51.  Ideal  Accuracy. — The  average  student  is  liable  to 
have  more  or  less  difficulty  in  grasping  the  idea  that 
accuracy  is  always  a  relative  matter  and  absolute  precision 
of  measurement   is   an  impossibility.     This  is   usually 


IV 


SIGNIFICANT  FIGURES 


57 


because  he  has  had  very  little  practice  in  careful  measure- 
ment and  at  the  same  time  his  previous  study  of  arith- 
metic has  emphasized  a  condition  of  infinite  accuracy  of 
numerical  values.  Such  a  number  as  12.5  has  been 
supposed  not  only  to  mean  the  same  thing  as  12.50  but 
also  to  be  equal  to  12.500000 ...  to  an  unlimited  number 
of  decimal  places.  This  is  quite  proper  and  satisfactory 
as  long  as  one  realizes  that  he  is  dealing  with  imaginary 
quantities,  or  perhaps  it  would  be  better  to  speak  of  them 
as  ideal  quantities,  perfections  of  measurement  which 
have  no  more  reality  of  existence  than  the  point,  line, 
plane,  or  cube,  of  geom- 
etry. The  smoothest  sur- 
face of  a  table  does  not 
come  as  near  to  being  a 
plane  as  does  the  surface 
of  an  " optically  worked" 
block  of  glass  or  a  "  Whit- 
worth  plane,"  and  even 
the  smoothest  possible  sur- 
face  can  be  magnified  so 
as  to  show  that  it  contains 
irregularities  everywhere. 
Perhaps  if  it  were  magni- 
fied  enough  we  could  see 
that  its  shape  would  not 
even  remain  constant,  but 
individual  molecules  would 
be  found  swinging  back 
and  forth  or  possibly  es- 


(3 


FIG  .  1 1 .  DIAGRAMMATIC  CROSS- 
SECTION. — A  metal  cube,  greatly 
magnified,  to  show  that  there  is  no 
plane  surface  of  contact  between 
the  metal  and  the  air  above  it. 


caping  from  the  surface. 
A  geometrical    plane  cer- 
tainly corresponds  to  nothing  in  reality,  and  perfect  ac- 
curacy of  number  is  just  as  much  an  imaginary  concept. 


58  THEORY  OF  MEASUREMENTS  §52 

52.  Decimal  Accuracy. — If  12.5  cm.,  as  a  measurement, 
does  not  mean  the  ideal  number  12.500000000.  .  .  to  an 
infinite  number  of  decimal  places  what  does  it  mean? 
As  different  measurements  are  likely  to  be  made  with 
different  degrees  of  accuracy  the  universally  adopted 
convention  is  merely  the  common-sense  one  that  the 
statement  of  a  measurement  must  be  accurate  as  far  as  it 
goes;  and  it  should  go  far  enough  to  express  the  accuracy 
of  the  determination.  Thus  "12.8  cm."  means  a  length 
that  is  nearer  to  precisely  12.800.  .  .  than  to  precisely 
12.7  or  12.9  cm.,  i.  e.,  that  its  "rounded-off  "  value  would 
be  12.8  cm.,  not  12.7  or  12.9.  If  a  length  is  written 
"12.80  cm.,"  however,  this  implies  that  the  stated  meas- 
urement is  nearer  to  this  same  precise  12.8  or  12.80  or 
12.800000.  .  .  than  it  is  to  either  12.79  or  12.81  cm.,  in 
other  words,  that  it  has  been  measured  to  hundredths 
of  a  centimetre  and  found  to  be  between  12.79J  and 
12.80|  cm.,  so  that  it  can  properly  be  rounded  off  .to 
12.80.  The  other  description,  "12.8  cm.,"  means 
between  12.7|  and  12. 8J;  it  states  nothing  about  hun- 
dredths of  a  centimetre,  and  can  correctly  represent  any 
lengths  between  the  limits  just  given;  for  example 
12.75,  12.76,  12.77,  12.78,  12.79,  12.80,  12.81,  12.82, 
12.83,  12.84,  or  12.85,  for  each  one  of  these  could  be 
rounded  off  to  12.8.  To  write  such  a  length  of  12.8  cm. 
in  the  form  "12.80  cm."  would  be  to  violate  the  rule 
that  a  statement  should  be  accurate  as  far  as  it  goes, 
for  it  would  go  as  far  as  hundredths  (stating  that  there 
were  eighty  of  them),  and  the  chances  are  ten  to  one 
that  it  would  be  one  of  the  other  numbers  of  hundredths 
given  above.  On  the  other  hand,  if  an  observer  de- 
termined a  length  to  be  12.80  cm.,  that  is,  if  he  measured 
the  length  as  12  cm.  +  8  tenths  +  0  hundredths — if 
he  looked  for  hundredths  and  established  the  fact  that 


IV  SIGNIFICANT  FIGURES  59 

there  were  none  of  them, — then  to  state  the  measurement 
only  as  12.8  cm.  would  not  be  doing  justice  to  his  own 
accuracy,  for  he  would  imply  that  the  correct  number  of 
tenths  was  merely  known  to  be  nearer  8  than  7,  namely 
greater  than  7.5,  whereas  he  had  already  found  it  to  be 
nearer  8  than  7.9,  namely  greater  than  7.95. 

When  a  carpenter  says  "just  8  inches"  he  probably 
means  "nearer  to  8f  than  to  8|  or  7f  inches,"  a  sixteenth 
of  an  inch  one  way  or  the  other  being  unimportant. 
When  a  machinist  says  "just  8  inches"  he  may  mean 
"nearer  to  8-fa  than  to  7ff  or  to  8^¥,"  a  half-sixty-fourth 
or  hundred-and-twenty-eighth  of  an  inch  being  negligible 
to  him.  When  another  person  says  "just  8  inches"  we 
must  know  what  kind  of  materials  he  works  with  before 
we  can  tell  the  meaning  of  his  word  "just."  If  decimal 
subdivisions  were  everywhere  used  the  carpenter's  eight 
inches  would  probably  mean  8.0  while  the  machinist's 
would  mean  8.00;  for  one  man  "8"  would  mean  "between 
7.950  and  8.050"  while  for  the  other  it  would  mean 
"between  7.995  and  8.005."  It  is  for  the  sake  of  avoiding 
such  ambiguities  that  the  scientist  has  adopted  the  rule 
that  "8"  means  "between  7.500  and  8.500";  "8.0" 
means  "between  7.950  and  8.050";  "8.00"  means 
"between  7.995  and  8.005";  "8.000"  means  "between 
7.999J  and  8.000J";  etc.;  in  other  words :  no  more  figures 
are  to  be  written  down  than  are  known  to  be  correct;  and, 
no  figures  that  are  known  to  be  correct  should  be  omitted. 
This  principle  is  simple  enough  when  it  has  once  been 
properly  comprehended,  and  after  that  there  is  not  much 
danger  of  the  student's  "rounding  off"  a  carefully 
obtained  measurement  like  2.836  gm.  to  2.84  gm.  merely 
for  the  sake  of  doing  some  rounding  off.  There  is  a  very 
decided  likelihood,  however,  that  he  will  often  forget  to 
write  down  a  final  significant  zero;  if  two  lengths  are 


60  THEORY  OF  MEASUREMENTS  §53 

147  mm.  and  160  mm.  the  tendency  when  writing  them 
in  centimetres  is  to  put  down  14.7  and  16.  If  the  zero 
is  as  important  as  the  seven  when  writing  millimetres 
the  same  is  equally  true  when  writing  centimetres. 
Suppose  the  diameter  of  the  ?r-disc  is  found  to  be  "  just  8" 
centimetres;  the  measurement  should  be  stated  as  8.00 
cm.  if  tenths  of  a  millimetre  were  estimated  and  none 
were  found,  but  it  should  be  given  as  8.0  cm.  if  the  student 
read  the  millimetres  but  was  unable  to  make  an  estimate 
of  smaller  amounts.  There  is  nothing  to  show  which 
degree  of  accuracy  was  obtained  if  the  diameter  is  put 
down  as  8  cm.  " because  it  came  out  just  even." 

53.  Significant  Figures. — In  the  expression  6.2  cm. 
both  the  figure  6  and  the  figure  2  mean  something  or  are 
significant.  In  the  expression  62  mm.  there  are  likewise 
two  significant  figures.  When  the  same  length  is  written 
in  the  form  .062  metre  there  is  no  difference  in  what  is 
signified,  and  although  the  number  has  three  figures  it  is 
still  said  to  have  only  two  significant  figures;  the  zero  is 
present  merely  for  the  purpose  of  showing  which  decimal 
places  are  occupied  by  the  six  and  the  two,  or,  in  other 
words,  for  fixing  the  location  of  the  decimal  point.  If 
the  same  length  is  called  62  thousands  of  micra  or 
62000  ju  there  are  still  only  two  significant  figures,  and 
again  the  ciphers  serve  only  to  show  that  the  six  is 
located  in  tens-of-thousands'  place,  i.  e.,  to  fix  the 
position  of  the  decimal  point.  In  arable  notation,  the 
figures  of  which  a  number  is  composed,  except  for  one  or 
more  consecutive  ciphers  placed  at  its  beginning  or  end  for 
the  purpose  of  locating  the  decimal  point,  are  called  its 
significant  figures.  In  accordance  with  this  definition  it 
will  be  clear  that  only  two  of  the  three  figures  of  0.75 
gm.  are  significant,  and  only  one  of  the  two  figures  of 
such  an  expression  as  05c.,  in  which  a  superfluous  zero 


IV  SIGNIFICANT  FIGURES  61 

is  sometimes  written.  Non-significant  ciphers  occur 
sometimes  on  the  left,  as  in  the  statement  that  a  certain 
light-wave  has  a  length  of  .00005086  cm.,  and  sometimes 
on  the  right,  as  when  the  sun  is  stated  to  be  93000000 
miles  from  the  earth;  the  first  of  these  numbers  has  four 
significant  figures,  the  second  has  only  two.  It  will  be 
noticed  that  a  number  like  the  last  causes  trouble  in 
applying  the  rule  of  making  it  "  accurate  as  far  as  it 
goes,"  for  only  the  first  two  or  three  figures  are  known, 
but  eight  are  needed  in  order  to  place  the  decimal  point. 
A  further  source  of  trouble  lies  in  the  fact  that  the  last 
figure  which  is  significant  may  happen  to  be  a  cipher 
instead  of  some  other  digit.  If  this  is  to  the  right  of  the 
decimal  point  the  zero  is  of  course  written  when  significant 
and  omitted  when  not  (as  explained  in  §  52),  but  what 
is  to  be  done  when  a  similar  case  occurs  in  which  the 
significant  zero  occurs  to  the  left  of  the  decimal  point? 
If  a  building  is  said  to  be  worth  fourteen  thousand 
dollars  how  is  any  one  to  tell  whether  this  means  14 
thousands  of  dollars,  i.  e.,  nearer  to  14  than  to  13  or  15 
of  these  thousands,  or  whether  it  means  exactly  14000 
dollars  and  no  cents,  or  whether  the  number  of  significant 
figures  is  not  intended  to  be  either  two  or  seven  but  some 
intermediate  number?  Suppose  the  number  of  millions 
of  miles  from  the  earth  to  the  sun  at  some  particular 
time  is  found  to  be  93.00;  we  need  two  different  symbols 
for  our  ciphers  so  that  we  can  write  93,00o,ooo  miles 
to  show  that  the  first  two  ciphers  are  significant  while 
the  last  four  are  not.  There  is  really  no  reason  for 
using  ciphers  at  all  in  the  last  four  places,  except  that  it 
is  customary,  and  it  would  be  better  to  use  some  other 
character,  such  as  g,  and  write  9300gggg.  Neither  of 
these  methods  is  ever  used,  however,  but  the  same  result 
is  achieved  by  a  notation  that  will  be  explained  later  on. 


62 


THEORY  OF  MEASUREMENTS 


§53 


The  length  of  an  inch  has  been  determined  to  be  be- 
tween 25.39977  and  25.39978  mm.  To  state  that  it  is 
25.40  mm.  is  correct,  because  this  value  is  accurate  as 
far  as  it  goes;  but  to  say  that  1  inch  =  25.39  mm.  would 
be  wrong,  for  the  true  number  of  hundredths  is  nearer  to 
40  than  to  39.  If  it  is  desirable  to  use  an  approximate 
value,  so  that  rounding  off  is  permissible,  how  many  of 
the  following  statements  are  correct  and  how  many  are 
positively  wrong?  1  in.  =  25.4  mm.;  1  in.  =  25.40  mm.; 
1  in.  =  25.400  mm.;  1  in.  =  25.4000  mm. 

Which  of  the  following  values  of  TT  are  correct,  and 
which  are  incorrect?  3.141592;  3.141593;  3.141600; 
3.14160;  3.1416;  3.1415;  3.142;  3.141;  3.15;  3.1; 


FIG.  12.  RELATIVE  AND  ABSOLUTE  ACCURACY. — The  large  rec- 
tangle comes  nearer  to  the  shape  of  a  perfect  square  than  the  small 
one  does,  although  the  difference  in  its  two  dimensions  is  precisely 
the  same  as  the  difference  between  the  height  and  the  width  of  the 
small  rectangle.  Accuracy  or  inaccuracy  can  be  considered  great 
or  small  only  relatively  to  the  size  of  the  quantity  that  is  being 
measured. 

3.142857   (=  22/7);   3.    (II  Chronicles,   chap.  4,  v.   2); 
3.16  (=  V10). 

At  ordinary  room  temperature  (20°  C.)  is  the  density 


IV  SIGNIFICANT  FIGURES  63 

of  water  equal  to  1?  Is  it  1.0?  1.00?  1.000?  To  what 
point  must  its  temperature  be  lowered  in  order  to  make 
its  density  1.0000?  (See  tables;  appendix.) 

54.  Relative  Accuracy. — The  diagram  (Fig.  12)  shows 
two  rectangles  which  are  approximately  squares.  The 
difference  between  the  height  and  the  width  of  the  larger 
one  is  just  the  same  as  the  difference  between  the  height 
and  the  width  of  the  smaller,  yet  the  small  rectangle  is 
obviously  a  less  accurate  approximation  to  the  shape  of  a 
perfect  square  than  is  the  large  one.  This  may  serve  as 
an  illustration  of  the  important  general  statement  that 
accuracy  is  a  matter  of  relative  amount  rather  than  of 
absolute  amount.  A  sixteenth  of  an  inch  has  the  same 
absolute  value  wherever  it  occurs,  but  it  is  a  considerable 
part  of  a  quarter-inch  length  while  it  is  relatively  insig- 
nificant in  comparison  with  a  whole  inch. 

The  relative  accuracy  of  a  measurement  accordingly 
depends  upon  two  things:  how  much  its  absolute  dif- 
ference from  the  truth  amounts  to,  and  how  large  the 
measurement  itself  is.  If  two  points  on  the  earth's 
surface  are  found  by  careful  surveying  to  be  10  miles 
apart  the  determination  of  distance  may  easily  be  in 
error  by  more  than  a  foot,  and  even  with  the  most  ex- 
tremely careful  triangulation  the  error  is  likely  to  be  as 
much  as  four  inches.  An  error  of  a  quarter  of  an  inch, 
however,  in  measuring  the  thickness  of  a  door  could 
hardly  be  made  even  with  the  clumsiest  of  measuring 
apparatus.  It  would  plainly  be  misleading  to  say  that  the 
clumsy  measurement  should  be  considered  more  accurate 
than  the  careful  one  on  account  of  i  inch  being  less  than 
4  inches.  The  only  consistent  way  of  looking  at  the 
matter  is  to  inquire  how  large  a  fraction  of  the  total 
measurement  the  error  amounts  to.  Suppose  the  thick- 
ness of  the  door  is  1J  inches;  how  large  a  part  of  this 


64  THEORY  OF  MEASUREMENTS  §55 

measurement  is  the  error  of  J  inch?  Obviously  it  is  one 
sixth  of  the  total  or  an  error  of  more  than  16  percent; 
while  four  inches  out  of  a  total  of  ten  miles  is  not  nearly 
a  sixth,  but  is  roughly  an  error  of  one  out  of  a  hundred 
and  fifty  thousand,  or  about  six  per  million,  or  about 
.0006  of  1  percent. 

How  large  an  error  is  an  eighth  of  an  inch  when  half  of 
an  inch  is  being  measured  (Fig.  12)?  How  large  is  a 
sixteenth  of  an  inch  out  of  a  total  of  an  inch  and  three 
quarters? 

If  a  measurement  is  stated  to  be  12.8  cm.  when  it  is 
known  to  be  between  12.750  and  12.850  cm.  what  is  the 
greatest  possible  error  of  the  statement?  Answer:  .05 
out  of  a  total  of  12.75.  This  is  the  same  as  05.  out  of 
1275.,  or  5  per  1275,  or  1  per  255;  1/250  would  be  4/1000 
or  .004,  so  1/255  must  be  a  little  less  than  .004,  i.  e.j  a 
little  less  than  0.4  percent. 

55.  Calculation  of  Relative  Errors. — The  relative  error 
of  a  measurement  does  not  usually  need  to  be  calculated 
with  any  very  great  care.  Where  numbers  are  as  different 
as  6  in  tens-of-thousandths'  place  (10-mile  survey,  above) 
and  16  in  units'  place  (thickness  of  door)  the  location  of 
the  decimal  point  is  really  more  important  than  the  size 
of  the  significant  figure  that  occupies  either  place ;  to  call 
the  former  number  "a  few  ten-thousandths  of  a  per-cent" 
and  the  latter  "some  ten  or  twenty  percent"  gives  all 
the  information  that  is  needed.  This  means  that  a 
calculation  of  relative  error  never  needs  to  be  done  on 
paper  but  can  always  be  worked  out  as  a  rough  mental 
calculation.  Thus,  in  the  illustration  given  above, 
1  foot  is  1/5280  of  1  mile,  hence  it  is  1/52800  of  10  miles 
or  roughly  about  1/50,000;  and  4  inches,  being  f  of  1 
foot,  is  £/50,000  of  the  whole  distance,  or  1/150,000.  The 
denominator  of  this  fraction  is  about  a  sixth  of  a  million 


IV 


SIGNIFICANT  FIGURES 


65 


(since  15  is  about  |  of  100),  so  1/150,000  =  6/1,000,000 
=  .000006  =  .0006  percent. 

Decide  mentally  what  percent  .01  is  of  7.23.  Ans. : 
.0015,  or  .15  per  cent. 

What  percent  of  94.07  is  .01? 

56.  "  Decimal  Places  "  versus  "  Significant  Figures." 
— If  a  length  is  stated  as  174.2  cm.  the  inference  is  that  it 
is  nearer  to  that  exact  amount  than  to  174.1  or  174.3 
cm.,  namely  that  its  error  certainly  is  not  as  much  as 
0.1  out  of  174.2.  This  is  the  same  as  saying  that  it  is  not 
as  much  as  1  out  of  1742,  or  1  out  of  nearly  2000,  or  5  per 
10000,  or  .0005,  or  .05  percent. 

In  the  following  table  the  left-hand  column  contains 
five  numbers,  all  of  which  are  carried  out  to  the  same 
number  of  decimal  places;  namely, 
two.  In  the  right-hand  column  notice 
that  the  same  five  numbers  occur, 
but  each  one  of  them  is  carried  out  to 
the  same  number  (3)  of  significant 
figures,  no  matter  how  many  decimal 
places  there  may  be.  If  the  accuracy 
of  each  of  these  ten  numbers  should 
be  worked  out  in  the  way  that  has 
just  been  explained,  would  the  num- 
bers in  the  left-hand  column  turn  out 
to  be  all  of  approximately  equal  accu- 
racy while  the  right-hand  column 
showed  great  accuracy  for  one  num- 
ber and  little  for  another,  or  would  the 
numbers  on  the  right  be  the  ones  that 
would  be  about  equally  accurate  while 
those  on  the  left  fluctuated?  In  other 
words,  is  accuracy  a  matter  of  decimal  places  or  of  sig- 
nificant figures? 
6 


7.23 

94.07 

0.52 

428.00 

66.67 


7.23 
94.1 
0.522 

428. 
66.7 


TABLE  OF  FIVE 
NUMBERS. — In  the 
left-hand  column 
each  number  has 
two  decimal  places 
but  some  have  more 
significant  figures 
than  others.  In  the 
right-hand  column, 
each  number  has 
three  significant  fig- 
ures but  some  have 
more  decimal  places 
than  others. 


66 


THEORY  OF  MEASUREMENTS 


§57 


The  table  has  been  repeated  on  this  page  and  shows  the 
answer  to  the  question.  Beside  each  of  the  ten  numbers 
its  accuracy  has  been  written  down,  as  a  percentage,  and 
it  will  be  seen  that  the  numbers  in  the  right-hand  column 
all  show  about  the  same  degree  of  accuracy,  while  those 
in  the  left-hand  column  differ  widely. 


.1% 

7.23 

7.23 

.1% 

.01% 

94.07 

94.1 

.1% 

2.     % 

0.52 

0.522 

.2% 

.002% 

428.00 

428. 

.2% 

.01% 

66.67 

66.7 

.1% 

TABLE    SHOWING    ACCURACY    OF  NUMBERS. — Notice   that   the 

number  of  decimal  places  to  which  a  measurement  is  carried  out 

has  nothing  to  do  with  its  accuracy.  It  is  the  number  of  significant 
figures  that  determines  the  matter. 

Turn  to  the  table  in  the  appendix  where  errors  are 
classified  according  to  their  size  and  write  the  appropriate 
word  opposite  each  of  the  percentages  given  in  the 
right-hand  column  of  the  table  on  this  page.  Then  do 
the  same  way  with  those  in  the  left-hand  column. 

57.  Rule  for  the  Relative  Difference  of  Two  Measure- 
ments.— The  difference  between  3.11  and  TT  (=  3.14)  is 
3  out  of  314,  and  the  difference  between  3.17  and  TT  is 
likewise  3/314,  not  3/317;  i.  e.,  the  numerical  error  is  to 
be  divided  by  the  true  or  theoretical  value  rather  than 
by  the  experimental  or  erroneous  value.  It  is  often 
desirable,  however,  to  compare  two  values  which  are 
equally  good,  according  to  one's  available  knowledge  of 
them.  When  there  is  no  standard  and  no  reason  for 
choosing  one  of  the  measurements  rather  than  the  other 
the  accepted  procedure  is  to  divide  the  difference  by  the 
greater  value.  For  example,  the  numbers  4  and  5  would 
be  said  to  differ  from  each  other  by  20  percent,  not  25 
percent,  for  the  difference  divided  by  the  greater  number 


IV  SIGNIFICANT  FIGURES  67 

is  one  fifth,  not  one  fourth.  In  cases  of  fairly  accurate 
measurements  it  is  unimportant  whether  the  larger 
number  or  the  smaller  one  is  taken  for  the  divisor,  but 
for  the  sake  of  uniformity  it  is  customary  to  choose  the 
larger  one. 

Apply  this  rule  to  your  two  measurements  of  an  ir- 
regular area  (§  26),  obtained  by  counting  squares  and 
by  constructing  geometrical  figures.  How  much  rel- 
ative difference  is  there  in  the  results  of  the  two  methods? 

58.  Accuracy  of  a  Calculated  Result. — Multiply  65.97 
by  24.15,  using  the  abridged  method  of  multiplication. 
Compare  your  product  with  that  of  example  (c),  §  16. 
It  will  be  noticed  that  changing  the  fourth  figure  of  one 
factor  has  produced  a  change  in  the  fourth  figure  of  the 
product.     This  means  that  if  only  three  figures  of  the 
factor  had  been  known,  the  fourth  being  uncertain,  no 
calculation   could   have   given  more  than  three  trust- 
worthy figures  of  the  product,  because  the  fourth  figure 
would  have  depended  upon  the  unknown  fourth  figure  of 
one  of  the  factors.    Likewise,  in  division,  if  only  five  figures 
are  known  of  either  the  divisor  or  the  dividend  there  is 
nothing  to  be  gained  by  keeping  more  than  five  figures 
in  the  other  one;  only  five  figures  of  the  quotient  will 
have  any  meaning,  and  if  further  figures  are  obtained 
by  any  process  of  calculation  they  will  be  unjustified 
and  misleading.     The  general  rule  will  be  obvious:  the 
result  of  a  multiplication  or  division  will  have  no  greater 
accuracy  than  that  of  the  least  accurate  of  the  data 
from  which  it  is  obtained. 

59.  Accuracy  of  the  Abridged  Methods. — Remember- 
ing that  the  accuracy  with  which  a  quantity  is  expressed 
depends  not  upon  the  number  of  decimal  places  but  upon 
the  number  of  significant  figures  and  keeping  in  mind 
the  fact  that  the  number  of  trustworthy  figures  in  a 


68  THEORY  OF  MEASUREMENTS  §60 

product  is  the  same  as  the  number  in  its  least  accurate 
factor,  turn  back  to  your  notes  on  §§  15  and  16  and  ob- 
serve that  the  method  of  abridged  division  automatically 
gives  just  the  number  of  figures  in  the  quotient  that  are 
needed  if  no  figures  of  the  dividend  are  " brought  down"; 
and  that  abridged  multiplication  always  gives  at  least 
as  many  as  are  in  the  shortest  factor.  It  will  not  give 
any  superfluous  figures  if  the  longer  factor  is  used  as  the 
multiplier,  but  will  give  as  many  as  the  longer  factor 
contains  if  that  is  used  as  the  multiplicand.  Of  course 
the  best  method  is  to  round  off  the  longer  number  before 
beginning  the  calculation,  so  that  it  has  no  more  figures 
than  the  shorter  one. 

60.  Standard  Form. — To  avoid  a  long  string  of  figures 
when  writing  very  large  or  very  small  numbers  it  is 
customary  to  divide  a  number  into  two  factors,  one  of 
them  being  a  power  of  ten.  Thus,  .00000017  and 
632000000000  are  the  same  as  17  X  10~8  and  632  X  109 
respectively.  -This  notation  also  makes  it  possible  to 
write  93000000  unequivocally  with  either  two  significant 
figures  or  four,  as  may  be  desired  (see  §  53) ,  for  it  can  be 
put  either  in  the  form  9.3  X  107  or  in  the  form  9.300 
X  107.  The  same  value  and  accuracy  for  9.300  X  107 
would  be  retained  just-  as  well  by  writing  93.00  X  106 
or  930.0  X  105,  but  it  is  customary  to  choose  the  power 
of  ten  so  that  the  other  factor  shall  have  just  one  sig- 
nificant figure  to  the  left  of  the  decimal  point.  The 
number  is  then  said  to  be  written  in  standard  form. 

Write  the  following  numbers  in  standard  form: 
2946.3;  632  X  109  (ans.:  6.32  X  1011);  17  X  10~8; 
25.39978;  0.0073;  .007300;  666.6;  .001;  .0010;  107.42; 
186000;  2.5400;  3.1416;  9.9942;  2.54gg.  Prove  that 
each  result  is  correct  by  performing  the  indicated  mul- 
tiplication. 


IV  SIGNIFICANT  FIGURES  69 

Write  a  definition  of  standard  form  in  your  own  words. 

61.  Questions  and  Exercises. — 1.  What  precautions 
did  you  take  in  order  to  measure  the  diameter  of  the  brass 
disc  with  the  greatest  possible  accuracy? 

2.  Write  your  measured  value  of  ?r,  carrying  it  out  just 
far  enough  to  show  one  wrong  figure.     How  many  sig- 
nificant figures  of  3.141625  are  correct?     Of  3.1424? 

3.  State  some  of  the  possible  causes  that  make  your 
determination  of  TT  incorrect.     Would  there  be  any  ad- 
vantage in  taking  the  average  of  several  measurements  of 
the  circumference?     In  rolling  the  disc  through  two  or 
more  consecutive  revolutions  and  measuring  the  total 
distance? 

4.  Is  there  any  difference  in  meaning  between  the 
italicized  statement  at  the  beginning  of  §  52  and  the 
one  near  the  end  of  the  same  section?     If  so,  what? 

5.  How  many  significant  figures  do  you  think  there  are 
in  the  length  of  the  earth's  quadrant  as  given  in  §  19? 

6.  Show  that  the  last  statement  in  §  57  is,  correct  by 
taking  some  measurement  that  you  have  made  as  an 
example. 

7.  If  one  gram  is  equal  to  15.432  grains  how  much  is 
five  grams?     If  a  weight  of  five  grams  is  the  same  as 
77.16  grains  how  much   is  one   gram?     (The   answers 
77.16  grains  and  15.432  grains  are  both  wrong.) 

8.  How  is  it  that  a  metre  can  be  measured  more  ac- 
curately (§  19)  than  a  centimetre? 

9.  Each  side  of  a  square  measures  82.5  mm.     How 
many  centimetres  long  is  its  entire  periphery?     ("  Just  33 
cm."  is  not  the  correct  answer.) 

10.  Use  the  abridged  method  for  multiplying  12  by 
13,    and    for    multiplying    13    by    12.     Why    does   the 
answer   come   out    16   each  time?     How  should  it   be 
pointed  off?     How  many  of  its  figures  are  significant? 


70  THEORY  OF  MEASUREMENTS  §61 

If  you  wanted  to  obtain  three  significant  figures  in  the 
product,  using  the  abridged  method,  what  would  you  do? 
(Answer:  Use  three  significant  figures  for  both  factors, 
12.0  and  13.0.  Try  it.) 

11.  In  exercise  9  how  many  significant  figures  did  you 
keep  in  the  product  of  82.5  X  4?     How  many  are  you 
entitled  to  keep?     Does  the  82.5  mean  the  same  as  82.50? 
Does  the  4  mean  the  same  as  4.0?     As  4.000? 

12.  Turn  to  the  table  in  the  appendix  where  the  density 
of  water  is  given.     Does  water  at  a  temperature  of  4°  C. 
have    a    density    of    1?     Of    1.000?     Of    1.0000?     Of 
1.00000?     Correct  the  following  statement  by  crossing 
off  the  unjustifiable  figures,  but  keep  all  that  are  correct: 
"at  ordinary  room  temperatures  water  has  density  of 
1.00000." 

13.  In  §  35  why  were  you  justified  in  adding  ciphers 
ad  libitum  to  the  number  180? 


V.     LOGARITHMS 

62.  Definitions. — The  logarithm  of  a  number  is  denned 
as  the  power  to  which  ten  or  some  other  numerical  quan- 
tity must  be  raised  in  order  to  give  the  specified  number. 
Thus,  the  logarithm  of  a  thousand  is  3  and  the  logarithm 
of  a  hundred  is  2,  for  1000  =  103  and  100  =  102.     These 
statements  are  usually  abbreviated  to  " log  1000  =  3" 
and   "log   100  =  2."     The  number  which  is  raised  to 
some  power  is  called  the  base,  and  logarithms  which 
have  10  for  a  base  are  called  common  logarithms.     For 
theoretical  purposes  what  are  known  as  natural  logarithms 
are  often  used;  their  base  is  a  number  which  is  denoted 
by  the  letter  e  (approximately  2.71828)  and  is  equal  to 
the  infinite  series  1  +  1  +  1/2+1/2 -3  +  1/2 -3 -4+1/2 -3 -4 -5 
+  .  .  .  or  to  the  limit  of  [(1  +  (l/n)]n  when  n  is  increased 
indefinitely.     To  avoid  confusion  the  base  is  often  written 
as  a  subscript;  thus,  logic  100  =  2  and  loge  100  =  4.6052 
mean  exactly  the  same  thing  as  102  =  100  and  e4-6052 
=  100.     The  only  logarithms  that  will  be  considered  here 
are  those  whose  base  is  10. 

63.  Fundamental  Properties  of  Logarithms. — The  table 
on  the  next  page  gives  the  values  of  various  integral 
powers  of  ten;  in  other  words  it  gives  the  numbers  which 
have  integers  for  their  logarithms.     Pick  out  any  two 
logarithms    (exponents)    and   add   them.     Then   notice 
that  the  sum  which  is  thus  obtained  is  another  logarithm, 
namely,  the  logarithm  of  the  product  of  the  two  numbers 
that  correspond  to  the  original  logarithms.     For  example 
the  logarithm  of  100  is  2,  and  of  a  thousand  is  3 ;  adding 
them,  5  will  be  found  from  the  table  to  be  the  logarithm, 
not   of   the   sum   of    100  +  1000,    but   of   the   product 

71 


72 


THEORY  OF  MEASUREMENTS 


§63 


100  X  1000,  or  100000.  One  of  the  chief  uses  of  loga- 
rithms is  to  enable  a  multiplication 
to  be  performed  by  the  simpler 
process  of  addition.  In  the  par- 
ticular case  just  given  it  is  as  easy 
to  multiply  100  by  1000  directly  as 
it  is  to  add  their  logarithms  and 
see  what  number  corresponds  to 
the  sum,  but  an  exercise  like  6.28 
X  17.35  is  as  easy  as  100  X  1000 
when  worked  out  by  logarithms 
although  it  would  mean  much  more 
time  and  trouble  to  multiply  it  out, 
even  if  the  abridged  method  were 
used.  The  process  is  simply  to  add 
log  6.28  to  log  17.35  and  the  result 
will  be  the  logarithm  of  their  prod- 
In  general, 

log  a  +  log  b  =  log  (a  X  6).  (1) 

Try  numerical  values  for  the  following  also,  taking 
each  equation  separately  in  turn,  and  extending  the 
above  table  if  necessary: 

log  a  -  log  b  =  log  (a  -I-  6),        (2) 


106  =  1000000 
105  =  100000 
104  =  10000 
103  =  1000 
102  =  100 
101  =  10 
10°  =  1 

io-j  =  .1 

1C-2  =   .01 

10-3  =    <OQ1 

TABLE  OF  COMMON 
LOGARITHMS.  —  The 
exponent  of  the  base, 
10,  on  the  left,  is  called 
the  logarithm  of  the 
'  natural  number  '  on 
the  right. 

uct,  6.28  X  17.35. 


n  X  log  a  =  log  an, 
(log  a)  ^  n  =  log  ">/a. 


(3) 
(4) 


These  four  equations  give  the  fundamental  principles 
involved  in  the  use  of  logarithms.  The  student  should 
not  attempt  to  memorize  them  as  equations,  but  will  need 
to  be  perfectly  familiar  with  the  ideas  that  they  express. 
Notice  that  when  using  logarithms  addition  takes  the 
place  of  multiplication,  subtraction  of  division,  multi- 


LOGARITHMS 


73 


(from  equation  3) 


plication  of  raising  to  a  power,  and  division  of  root  ex- 
traction. Addition  and  subtraction  are  performed  on 
logarithms;  multiplication  and  division  are  also  performed 
upon  logarithms  but  the  multiplier  or  divisor  is  the 
number  itself  (natural  number,  as  it  is  often  called  to  dis- 
tinguish it  from  the  logarithm  or  logarithmic  number} , 
not  the  logarithm  of  the  number.  The  result  in  all 
cases  is  a  logarithm,  and  from  this  the  required  number 
is  found  by  consulting  a  table. 

64.  Common  logarithms. — The    advantage    of   using 
10  as  a  base  is  that 

log  (10  X  a)  =  log  10  +  log  a  (from  eq.  1) 

1  +  log  a 
and  in  general 

log  (10n  X  a)  =  n  log  10  +  log  a 
=  n  -h  log  a; 

for  example,  log  365  =  log  (102  nat.  no. 
X  3.65)  =  2  +  log  3.65.  Accordingly 
tables  of  common  logarithms  are 
made  out  only  for  natural  numbers 
between  1  and  10,  the  logarithms  of  all 
other  numbers  being  self-evident  from 
these. 

If  the  logarithm  of  3.65  is  .562  what 
is  the  logarithm  of  3650?  Ans. :  3.562. 
What  is  log  365?  Log  36.5?  Log 
.365?  Ans.:  -  1  +  .562.  Log 
.0000365?  Ans.:  -  5  +  .562.  (Do 
not  simplify  these  binomial  forms. 
They  are  easier  to  use  if  left  as  they 
are.) 

The  logarithms  of  numbers   other 


nat.  no. 

log. 

1 

.000 

2 

.301 

3 

.477 

4 

.602 

5 

.699 

6 

.778 

7 

.845 

8 

-903 

9 

.954 

10 

1.000 

TABLE  OF  LOG- 
ARITHMS. —  The 
logarithms  of  the 
natural  numbers 
from  1  to  10  are 
given  here  as  far  as 
the  first  three  deci- 
mal places. 


than  powers  of  10  are  in  general  incommensurable  and 


74  THEORY  OF  MEASUREMENTS  §65 

are  given  only  approximately  in  tables.  Use  only  the 
small  table  given  on  the  last  page  for  the  exercises  in 
the  following  paragraph. 

Find  2X3.  Answer:  log  2  =  .301;  log  3  -  .477; 
their  sum  is  .778,  and  by  looking  in  the  table  this  is 
found  to  be  the  logarithm  that  corresponds  to  the 
number  6.  (Six  is  sometimes  said  to  be  the  anti-loga- 
rithm of  .778.)  Using  logarithms,  find  2X4.  (Do  not 
perform  the  multiplication  mentally  and  then  look  for 
the  logarithm  of  8  to  verify  the  sum  of  log  2  and  log  4, 
but  consider  the  product  as  being  unknown  until  after 
you  have  been  directly  led  to  it  by  following  out  the 
logarithmic  process.)  Find  22;  find  32.  Find  4X5; 
V  9;  5  X  6;  50  X  6;  500  X  600. 

Calculate  the  value  of  e  from  the  infinite  series  given 
above. 

65.  Practical  Logarithm  Tables. — Examine  the  four- 
place  logarithm  table  in  the  appendix  at  the  end  of  this 
book,  and  notice  that  it  contains  the  same  succession  of 
numbers,  from  1  to  9,  as  the  small  table  wrhich  has  just 
been  used.  It  also  contains  the  same  succession  of 
logarithms,  from  .0  to  .'9;  but  the  intermediate  values, 
both  of  logarithms  (.0000  to  .9999)  and  of  natural 
numbers  (1.00  to  9.99),  are  given  at  smaller  intervals, 
and  without  any  decimal  points.  Verify  each  of  the 
following  statements  by  finding  the  required  logarithm 
in  line  with  the  first  two  figures  of  the  natural  number  as 
they  occur  in  the  left-hand  column,  and  in  the  column 
that  is  headed  by  the  third  figure:  log  3.65  =  .5623; 
log  3.66  =  .5635;  log  4.06  =  .6085;  log  7.70  =  .8865; 
log  77.0  =  1.8865  (§  64);  log  7700  =  3.8865;  log  .00077 
=  -  4  +  .8865. 

Find  the  logarithms  of  5.02;  5.01;  5.00;  50.0;  5000000. 

It  will  have  been  noticed  that  the  decimal  part  of 


V  LOGARITHMS  75 

a  logarithm  (sometimes  called  the  mantissa)  is  dependent 
only  upon  the  arrangement  of  signicant  figures  in  the 
natural  number;  e.  g.,  log  36500  =  4.5623;  log  .0365 
=  —  2  +  .5623.  The  integral  part  (sometimes  called 
the  characteristic  of  the  logarithm),  however,  is  deter- 
mined only  by  the  position  of  the  first  significant  figure; 
for  example,  for  any  number  beginning  in  tens-of- 
thousands'  place  it  is  4,  and  for  any  number  beginning  in 
hundredths'  place  it  is  —  2.  In  order  to  save  space  a 
number  like  log  .0365  is  customarily  written  in  the  form 
2.5623,  the  minus  sign  being  written  over  the  character- 
istic to  indicate  that  it  applies  only  to  the  whole  number 
while  the  decimal  part  is  always  positive. 

Write  -  2.60  in  logarithmic  form.  Ans.:  -  2.60  = 
-  2  -  .60  =  -  3  +  .40  =  3.40.  _ Write  -  1.4377  with 
the  decimal  part  positive.  Ans. :  2.5623. 

Write  the  characteristic  of  the  logarithm  of  each  of  the 
following:  5441;  27;  79264;  264;  73;  0.73;  0.073;  0.000073. 
Make  up  a  rule  for  finding  the  characteristic  of  the 
logarithm  of  any  number,  and  write  it  in  your  notebook. 

Write  the  logarithms  of  984;  982;  981;  980;  98;  9.8; 
.98;  .098;  7;  14.  Add  the  last  two  and  find  their  sum  in 
the  body  of  the  table;  see  what  number  in  the  margin 
corresponds  to  it,  and  verify  the  result  by  multiplying 
14  by  7. 

66.  Use  of  the  Table. — A  four-place  table  of  loga- 
rithms is  in  general  satisfactory  for  obtaining  the  logarithm 
of  a  number  that  has  four  significant  figures;  for  numbers 
of  three  significant  figures  it  is  not  necessary  to  keep 
more  than  three  decimal  places  of  the  logarithms;  for 
five-figure  accuracy  a  five-place  table  is  needed;  etc.  A 
four-place  table  is  generally  made  more  compact  by 
including  only  three-figure  values  for  the  natural  num- 
bers, and  when  the  logarithm  of  a  four-figure  value  is 


76  THEORY  OF  MEASUREMENTS  §66 

required  it  is  found  by  a  process  called  interpolation,  in 
which  it  is  assumed  that  small  differences  between  loga- 
rithms are  proportional  to  the  corresponding  differences 
in  their  antilogarithms.  Turn  to  the  table  and  notice 
that  (log  633  —  log  632)  is  exactly  one  third  as  large  as 
(log  635  —  log  632) ;  and  of  course  the  differences  in  the 
numbers  633  —  632  and  635  —  632  are  in  the  same  ratio. 
Suppose  the  logarithm  of  3.142  is  required:  The  table 
gives  log  3.14  and  log  3.15.  The  required  number  3.142 
is  larger  than  3.140  but  smaller  than  3.150;  in  an  orderly 
scale  of  numbers  it  would  be  located  just  one  fifth  of  the 
way  from  3.14  to  3.15.  The  assumption  is,  accordingly, 
that  its  logarithm  likewise  is  situated  one  fifth  of  the  way 
vfrom  log  3.14  to  log  3.15.  Log  3.14  =  .4969;  log  3.15 
=  .4983.  One  fifth  of  the  distance  from  .4969  to  .4983 
is  obtained  by  first  finding  that  distance — subtracting; 
4983  —  4969  =  14; — then  adding  the  required  fraction 
of  it  to  the  lower  logarithm — 1/5  of  14  =  3  (the  nearest 
whole  number),  and  4969  +  3  =  4972;  /.  log  3.142 
=  .4972. 

Suppose  the  number  is  required  whose  logarithm  is 
.2752:  Turn  to  the  table  and  notice  that  the  nearest 
logarithms  are  2742  and  2765.  Their  difference  (called 
the  tabular  difference)  is  23,  and  the  given  logarithm  2752 
is  10  larger  than  2742;  i.  e.,  it  lies  10/23  of  the  way  from 
2742  to  2765.  Accordingly  the  required  antilogarithm 
will  be  10/23  of  the  way  from  one  marginal  number  (188) 
to  the  other  (189);  that  is,  it  will  be  188J&  °r  188.4.  As 
the  characteristic  of  the  given  logarithm  is  zero  this 
should  be  pointed  off  as  1.884. 

The  small  multiplication  tables  at  the  side  of  the 
logarithmic  table  enable  a  fraction  like  10/23  to  be 
reduced  to  tenths  mentally.  Find  the  table  headed  23 
and  notice  that  it  gives  one  tenth  of  23  =  2.3;  .2  of 


V  LOGARITHMS  77 

23  =  4.6;  etc.  The  number  nearest  to  10  is  9.2,  which 
stands  opposite  4;  accordingly  10  twenty-thirds  comes 
nearer  to  4  tenths  than  to  5  tenths.  Where  1/5  of  14 
was  required,  above,  the  small  tables  would  have  been 
used,  if  necessary,  by  finding  the  number  which  is 
opposite  2  tenths  in  the  fourteen  table. 

Find  log  2.718.  Ans.  0.4343.  Find  log  3.333;  log 
1.234;  log  12.34;  log  123.4;  log  8888;  log  .4343;  log  3449. 

67.  The  Probability  Function.  —  In  much  of  the  stu- 
dent's later  work  it  will  be  important  to  know  how  e~^ 
varies  when  x  is  given  different  numerical  values.  Be- 
fore substituting  any  particular  number  for  x  it  will  be 
advisable  to  proceed  as  follows:  Let  the  function  e~x2 
be  denoted  by  y,  then 


taking  logarithms  of  each  side  this  becomes 

log  y  =  -  x2  log  e, 
from  which  it  follows  that 

—  log  y  =  x2  log  e. 
Taking  logarithms  of  each  side  again 

log  (  -  log  y)  =  log  (z2)  +  log  (log  e) 
or 

log  (-  log  y)  =  log  (x2)  +  1.6378. 

In  the  last  equation  it  is  easy  to  find  the  value  of  y  that 
corresponds  to  any  given  value  of  x.  The  procedure  is 
first  to  calculate  x2  and  find  its  logarithm;  then  add 
T.6378.  The  result  is  stated  by  the  equation  to  be  the 
logarithm  of  an  expression  enclosed  in  parenthesis,  so  the 
numerical  value  of  that  expression  is  easily  found  by  the 
process  of  obtaining  an  antilogarithm.  Then,  when  the 


78 


THEORY  OF  MEASUREMENTS 


§68 


value  of  ( —  log  y)  has  been  obtained  it  is  easy  to  write 
the  value  of  (+  log  y),  and  from  the  value  of  log  y 
there  is  no  difficulty  in  finding  y  itself. 

When  such  a  determination  is  to  be  made  for  several 
different  values  of  x  it  is  convenient  to  arrange  the 
various  calculated  quantities  in  the  form  of  a  table. 
Using  the  last  of  the  equations  that  was  derived  from 
y  =  e~x2,  and  the  tables  of  squares  and  of  logarithms  in 
the  appendix,  find  in  succession  the  values  of  x2,  log  x2, 
log  x2  +  1.6378,  log  (-  log  y),  -  log  y,  log  y,  and  y, 
for  each  one  of  the  following  values  of  z:  0,  .2,  .4,  .6,  .8, 
1.0,  1.2,  .  .  .  2.8,  3.0;  4.0;  5.0.  On  the  next  unused  left- 
hand  page  of  your  notebook  tabulate  the  results  in 
columns  headed  x,  x2,  log  x2,  etc.,  carrying  each  line  clear 
across  the  table,  as  shown  in  the  illustration,  before 
beginning  the  next  line. 


X 

xz 

log  x2 

log  (z2) 
+  1.6378 

log  (  -  log  y) 

-logy 

+  logy 

9 

0 

0 

—    00 

—   00 

—   00 

0 

0 

1 

.2 

.04 

2.6021 

2.2399 

2.2399 

.0174 

1.9826 

.961 

.4 

.16 

"1.2041 

2.8419 

28419 

.0695 

1.9305 

.852 

.6 

.8 

.36 
.64 

1.5563 

1.1941 

1.1941 

.699 

TABLE  CONSTRUCTED  WHEN  FINDING  THE  VALUES  OF  e~*2. — It 
is  important  that  the  table  be  filled  out  line  by  line,  not  column  by 
column. 

Leave  the  opposite  right-hand  page  of  the  notebook 
vacant  until  directions  for  using  it  are  reached  in  another 
chapter. 

68.  Questions   and   Exercises. — 1.  Find   the    numer- 
ical values  of  (1+TV)10;  (1+yW100;  and 
Ans.:  2.6;  2.7;  2.7. 


V  LOGARITHMS  79 

2.  If  log  10001  =  4.00004343  find  the  value  of 


Ans.:  2.718. 

3.  Find  the  natural  logarithm  of  ?r.     Ans.:  By  defi- 
nition, ex  =  TT  (§  62).      Solving  this  equation  will  give 
x  =  log  7r/log  e  =  1.1447. 

4.  Use  the   algebraical  fact  that   10*10"  =  IQX+V  to 
prove  equation  1  of  §  63,  which  was  merely  stated  without 
proof. 

5.  Turn  back  to  the  numbers  that  you  have  written 
in  standard  form  (§60,  If  2)  and  write  the  characteristic 
of  the  logarithm  of  each  of  them.     What  relationship 
can  be  observed? 

6.  Prove  that  the  second  equation  of  §  67  must  be 
true  if  the  first  one  is  true. 

7.  Write  down  any  number  that  has  three  significant 
figures.     Find  its  logarithm  from  the  table,  subtract  it 
mentally  from  0  (=  log  1),  and  find  the  antilogarithm  in 
order  to  obtain  the  reciprocal  of  the  number  that  was 
written  down.     The  mantissa  is  all  that  is  needed  for 
each  logarithm,  as  the  answer  can  be  pointed  off  by 
inspection.     For  example,  suppose  the  reciprocal  of  TT  is 
required:  log  3.14  =  497;  working  from  left  to  right, 
subtract  each  figure  of  the  logarithm  from  9  except  the 
last  one,  which  is  to  be  subtracted  from  10;  result,  503; 
antilog  503  =  318;  pointed  off,  1/3.14  =  .318. 

Find  1  -f-  269  by  using  the  logarithm  table,  but  without 
writing  down  any  figures.  The  answer  must  be  approxi- 
mately .04. 

Find  the  reciprocal  of  a  four-figure  number  (for  ex- 
ample, TT  or  e)  in  the  same  way. 


VI.     SMALL   MAGNITUDES 

Apparatus. — Platform  balance;  set  of  gram  weights; 
set  of  avoirdupois  weights. 

69.  Approximate  Values. — If  the  sum  of  one  hundred 
dollars  is  put  out  at  3  percent  compound  interest  it  is 
said  to  "  amount  "  to  103  dollars  after  one  year,  about 
106  dollars  after  two  years,  about  109  dollars  after  three 
years,  etc.     The  exact  values  of  the  latter  amounts  are 
106.09  (the  square  of  1.03;  in  hundreds  of  dollars)  and 
109.2781  (the  cube  of  1.03) ;  but  if  the  principal  had  been 
one  single  dollar  the  amount  that  could  have  been  repaid 
after  two  years  or  three  years  would  necessarily  have 
been  $1.06  or  $1.09,  respectively,  on  account  of  the  true 
amounts  being  less  than  $1.06J  in  the  first  case,  and 
nearer  to  $1.09  than  to  $1.10  in  the  second.     The  fact 
that  in  United  States  money  fractions  of  a  cent  cannot  be 
used   (except  for  purposes  of  calculation)   corresponds 
exactly  to  the  metrological  fact  that  any  measurement 
can  be  carried  out  to  some  particular  degree  of  precision 
but  no  further. 

70.  Negligible  Magnitudes. — Suppose  a  ruler  gradu- 
ated in  centimetres  and  millimetres  is  used  to  measure 
the  side  of  a  square,  and  by  estimating  tenths  of  a  milli- 
metre the  length  is  found  to  be  2.87  cm.     The  mathe- 
matical square  of  this  quantity  is  8.2369  cm2,  but  it  has 
already  been  seen  (§  58)  that  only  three  figures  of  this 
area  can  be  trusted,  because  nothing  is  known  about  the 
fourth  figure  of  the  measurement  from  which  it  is  de- 
rived.    Accordingly,  the  square  is  said  to  have  an  area 
of  8.24  cm2.     Similarly,  if  one  side  of  a  square  measures 
1.03  cm.,  the  measurement  being  correct  to  tenths  of  a 

80 


VI  SMALL  MAGNITUDES  81 

millimetre  but  nothing  being  known  about  hundredths 
of  a  millimetre,  then  its  area  will  be  correctly  expressed 
by  the  quantity  1.06  cm3,  and  the  volume  of  a  cube  that 
has  this  square  for  one  of  its  sides  will  be  1.09  cm3.  It 
should  be  noticed  (a)  that  with  an  ordinary  ruler  it  is 
impossible  to  measure  a  length  of  a  few  centimetres  with 
an  accuracy  greater  than  is  expressed  by  three  significant 
figures;  (6)  that  the  area  or  volume  calculated  from  such 
data  cannot  be  trusted  further  than  its  third  significant 
figure;  (c)  that  the  example  just  given  suggests  a  remark- 
ably simplified  process  of  calculation  where  some  quantity 
which  is  to  be  squared  or  cubed  is  a  little  greater  than  unity, 
namely:  (one  +  small  amount)2  =  one  +  twice  small  amt.} 
and  (one  +  small)3  =  1  +  3 (small). 

Decide  mentally,  by  induction,  the  value  of  (1.  — .02)2; 
then  prove  the  result  in  two  ways:  (a)  expanding  in 
accordance  with  the  binomial  theorem;  (b)  squaring  .98 
by  abridged  multiplication. 

The  justification  of  the  simplified  process  of  raising  a 
number  which  is  approximately  unity  to  a  power  will 
perhaps  be  made  more  evident  by  the  following  example : 

Suppose  that  a  metal  cube  has  been  constructed  ac- 
curately enough  to  measure  1.00000  cm.  along  each  edge. 
If  it  should  be  brought  from  a  cold  room  into  a  warm 
room  a  delicate  measuring  instrument  might  show  that 
the  change  of  temperature  had  increased  each  dimension 
to  1.00012  cm.  and  by  unabridged  multiplication  it  would 
be  easy  to  prove  that  the  area  of  each  side  was 
1.0002400144  cm2  and  that  the  volume  had  become 
1.000360043201728  cm3.  If  the  most  careful  measure- 
ments make  it  just  possible  to  distinguish  units  in  the 
fifth  decimal  place  then  tenths  of  those  units  (repre- 
sented by  the  sixth  decimal  place)  would  be  impossible 
to  measure,  and  the  attempt  to  state  not  only  tenths, 
7 


82  THEORY  OF  MEASUREMENTS  §71 

but  hundredths  and  thousandths  of  those  units  would 
be  absurd.  By  noticing  that  the  number  1.0002400144 
differs  from  the  value  obtained  by  abridged  multiplication 
(1.00024)  by  only  a  few  thousandths  of  the  smallest 
measurable  amount  we  can  see  clearly  why  the  area  of  a 
1.00012-cm.  square  is  and  must  be  1.00024  cm2. 
Similarly,  the  volume  of  the  cube  is  neither  more  nor 
less  than  1.00036  cm3,  and  the  string  of  figures  running 
out  ten  decimal  places  further  is  absolutely  meaningless. 

It  will  be  noticed  that  the  number  1.0002400144  is  in 
the  same  form  as  1  +  2x  +  z2,  the  square  of  (1  +  x), 
where  x  =  .00012;  also  that  1.000360043201728  cor- 
responds to  the  cube,  1  +  3x  +  3z2  +  x3.  This  is  an 
illustration  of  the  fact  that  when  dealing  with  objects 
of  the  real  world  which  is  evident  to  our  senses  it  may 
happen  that  a  measured  amount  is  so  small  that  its 
higher  powers,  algebraically  speaking,  are  relatively 
minute  beyond  all  perceptive  ability.  Of  course  this 
must  not  be  understood  as  meaning  that  the  cube  of  a 
measurable  length  can  ever  be  an  impalpable  volume; 
the  cube  of  1  +  x  is  even  a  larger  number,  1  +  3z;  it  is 
the  difference  in  size  between  this  "  physical  "  value, 
(1  +  x)s  =  1  +  3z,  and  the  true  mathematical  value, 
(1  +  x)3  =  1  +  3x  +  3z2  +  z3,  that  eludes  perception 
on  account  of  x2  being  extremely  small  in  comparison 
with  x,  which  is  itself  minute. 

71.  Formula  for  Powers. — The  examples  that  have 
been  given  above  suggest  that,  if  x  is  small  enough, 
(1  +  x)2  =  1  +  2x,  (1  +  x)3  =  1  +  3z,  and  in  general 
(1  +  x)n  =  1  +  nx.  The  matter  can  be  tested  by  mak- 
ing use  of  the  binomial  theorem: 

.   n(n  —  l)    .      n(n—l)(n  —  2)    , 
l±z)n  =  l  ±nx+  *2±  X    H ' 


VI 


SMALL  MAGNITUDES 


83 


This  shows  that  (1  d=  x)n  is  equal  to  1  db  nx  if  x  is  so 
small  that  x2,  x3,  etc.,  are  negligible,  the  only  possible 

1  X  S  \  S*  ^ 


i+S 


1  x  i 


FIG.  13.  THE  SQUARE  OF  1  +  5.  —  The  fact  that  6  is  small 
means  that  52  must  be  very  small.  The  error  produced  by  taking 
the  area  1  +  25  for  the  square  of  1  +  5  is  only  a  minute  fraction  of 
the  total  area. 

exception  being  in  case  n  should  be  so  large  that  it 
could  counterbalance  the  small  size  of  x  and  prevent  the 
term 

n(n  —  1) 


from  becoming  negligible.  In  the  practical  use  of  the 
formula,  however,  n  is  rarely  larger  than  2  or  3  while  x 
is  at  most  only  a  few  hundredths  and  is  usually  very 
much  smaller. 

72.  Properties  of  Deltas.  —  The  small  quantities  which 
have  been  considered  in  this  chapter  are  usually  sym- 
bolized by  the  Greek  letter  8.  It  has  been  seen  that 
an  important  property  of  deltas  is  given  by  the  equation 
(1  d=  5)n  =  1  ±  nd,  but  the  student  is  advised  to  learn 
this  in  the  form 

(1  +  6)»  =  1  +  n&,  (1) 


84  THEORY  OF  MEASUREMENTS  §72 

with  the  understanding  that  5  may  have  either  a  positive 
or  a  negative  value;  for  example  the  square  of  l  +  (  —  .03) 
is  equal  to  1  +  2(—  .03)  or  1  -  .06,  or  0.94. 

Find  the  value  of  each  of  the  following  expressions 
mentally:  .992;  .982;  .983;  .972;  1.000122;  1.000123; 
(1  -  .008)2;  .9923. 

The  ordinary  process  of  algebraical  division  shows  that 
1/(1  +  x)  =  1  -  x  +  x2  -  x3  +  z4  -  x5  +  -  •  •  .  From 
this  it  follows  that 


Divide  1  by  0.997.  Ans.:  .997  =  1.-.003;  !/(!-.  003) 
=  1  +  .003  =  1.003. 

Find  mentally  the  reciprocal  of  1.00012. 

Find  1/(1.00012)2  mentally  by  using  first  formula  (1) 
to  simplify  the  denominator  and  then  formula  (2)  to  clear 
of  fractions. 

Find  (1.00012)5  and  complete  the  following  formula 
for  yourself:  i/l  +  5  = 

If  a  number  is  so  small  that  its  square  is  negligible  it 
will  similarly  be  true  that  the  product  of  two  such 
numbers  will  be  negligible.  For  example,  if  a  number 
that  is  carried  out  to  thousandth's  place  is  1.007  its 
square  will  not  differ  from  1.014  by  a  single  thousandth: 
1.007  X  1.007  =  1.014049;  likewise  1.007  X  1.006  will 
not  differ  by  a  thousandth  from  1.013,  its  "  exact  " 
value  being  1.013042,  as  the  student  can  easily  prove 
by  considering  it  to  be  a  special  case  of  (1  +  x)(l  +  y) 
=  1  +  x  +  y  +  xy.  Accordingly, 

(1  +  50(1  +  52)  =  1  +  Si  +  52.  (3) 

This  equation  shows  the  advantage  of  keeping  the  signs 
positive  and  allowing  the  deltas  to  be  either  positive  or 


VI  SMALL  MAGNITUDES  85 

negative.  If  the  deltas  were  restricted  to  positive  values 
there  would  be  a  liability  to  error  unless  the  equation 
could  be  written  (1  ±  5i)(l  ±  52)  =  1  ±  5i  ±  dz. 

Find  mentally  1.04  X  0.98.  Ans.:  .98  =  1.  -  .02; 
(1  +  .04)  (1  -  .02)  =  1.02. 

Find  mentally  1.03  X  0.98;  also  1.00012  X  .99890 
and  prove  the  latter  by  abridged  multiplication. 

In  your  notebook  work  out  neatly  the  value  of  1.0021 
X  1.0037  in  three  different  ways:  (a)  by  unabridged  mul- 
plication;  (b)  by  abridged  multiplication;  (c)  by  the  use 
of  deltas,  writing  the  two  numbers  one  under  the  other 
and  adding  merely  the  deltas  on  paper  in  the  customary 
manner.  Compare  the  three  processes  carefully  and 
draw  the  moral  for  yourself. 

Remembering  that  a/6  is  the  same  as  a  X  (1/6)  write 
the  formula  for  (1  +  Si) /(I  +  <52). 

A  fact  that  is  useful  when  making  measurements  of 
mass  is  if  two  quantities  are  nearly  equal  to  each  other  the 
square  root  of  their  product  (or  geometrical  mean)  can  be 
obtained  by  taking  their  average  (or  arithmetical  mean). 
If  the  two  quantities  are  denoted  by  a  and  a  +  d  (to 
indicate  that  their  difference  is  a  small  magnitude)  this 
statement  can  be  proved  as  follows: 


l/a(a  +  5)  =  i/a  -  o(l  +  d/a) 


=  a  i/l  +  d/a  =  a(l  +  5/2a), 
since  d/a  is  also  a  small  magnitude.     But 
a(l  +  8/2a)  =  a(2a  +  d)/2a 


-  (4) 


If  an  object  appears  to  weigh  mi  when  placed  on  one 
of  the  scale  pans  of  a  balance,  and  m,z  when  on  the  other 


86  THEORY  OF  MEASUREMENTS  §72 

pan  it  can  be  proved  (see  Fig.  14)  that  its  true  mass  is 
I/ mi 7^2.  For  example,  a  metal  block  weighed  15.19  and 
15.23  gm.  on  the  two  sides  of  a  balance.  Its  true  weight 
is  accordingly  1/15.23  X  15.19  gm.  Equation  (4)  shows 
that  this  is  the  same  as  15.21  gm. 

1  1  +  8 


m, 

1+6 

~> 


FIG.  14.  DIAGRAM  OF  A  BALANCE. — The  force  multiplied  by  the 
distance  at  which  it  acts  is  the  same  on  each  side  .'.  a  =  mi(l+5);  also 
mz=x(l  +5).  Eliminating  (1+5)  gives  x/mi  =  m<Jx,  whence  x=  VwiW2. 

Weigh  your  whole  set  (16  ounces)  of  avoirdupois 
weights,  considered  as  an  unknown  mass  on  the  platform 
balance  against  the  brass  gram  weights.  Repeat  the 
process  on  the  other  pan  of  the  balance,  remembering 
that  the  reading  of  the  sliding  weight  is  additive  in 
one  case  and  subtractive  in  the  other.  Find  the  true 
weight,  in  grams,  of  the  avoirdupois  set. 

It  will  be  almost  self-evident  from  Fig.  15  that 

tan  8  ,     sin  5       -  ,~^ 

— —  =  1     and     —  =  1  (5) 

5  .  o 

if  it  is  remembered  that  an  angle  is  measured  by  the 
ratio  of  arc  to  radius. 

By  consulting  the  table  of  circular  functions  find  the 
largest  whole  number  of  degrees  for  which  tan  5  =  d 
to  four  decimal  places.  The  numerical  measure  of 


VI 


SMALL  MAGNITUDES 


87 


each  angle  will  be  found  in  line  with  the  number  of 
degrees,  but  in  the  column  headed  RAD,  an  abbreviation 
of  radian  measure,  which  is  another  synonym  for  circular 
measure. 


FIG.  15.  FUNCTIONS  OF  A  SMALL  ANGLE. — The  tangent  is  always 
larger  than  the  angle  and  the  sine  is  always  smaller,  but  the  ratio 
of  either  function  to  the  angle  comes  nearer  and  nearer  to  unity  as 
the  angle  is  made  smaller  and  smaller.  A  formal  proof  of  the  fact 
that  lim  sin  x/x  =  lim  x  =  lim  tan  x/x  can  easily  be  worked  out  if 
two  equal  angles  are  juxtaposed  so  that  their  sine  lines  complete  the 
chord  of  the  double  arc. 

If  an  accuracy  of  three  decimal  places  is  all  that  is 
needed  how  large  can  d  be  without  differing  from  tan  5? 
Ans. :  6°;  but  not  7°.  How  large  if  d  is  to  equal  sin  5? 

73.  Transformation  of  Operands. — It  will  be  noticed 
that  most  of  the  rules  that  have  been  given  for  operating 
upon  small  quantities  involve  functions  of  1  +  5,  not  of 
d  alone.  This  means  that  factoring,  or  some  other  proc- 
ess, is  often  necessary  in  order  to  put  an  expression 
intojthe  form  of  1  +  <5.  For  example,  V  50  is  equal  to 
1/72  +  1,  an  expression  which  is  not  in  the  form  of  a 
function  of  1  +  5  but  can  be  made  so  by  dividingjthe 
binomial  by  an  assumed  factor  49:  1/49  +  1 
=  1/49(1  +  1/49)  =  71/1+1/49  =  7(1  +  1/98).  Now 


88  THEORY  OF  MEASUREMENTS  §75 

it  will  be  noticed  that  98  is  a  little  less  than  100,  and  so 

•02j) 


-Wl  + ± 

-2/        \     r  100(1.-. I 

1  +  .02)  }  =  7(1.0102)  =  7.0714. 


o 

The  first  four  figures  of  this  result  are  correct.  Extreme 
accuracy  cannot  be  expected  where  a  delta  is  as  large  as 
one- or  two  percent;  in  most  physical  calculations  it  is 
much  smaller  than  this. 

74.  Recapitulation. — The  formula?  for  calculation 
with  small  quantities  are  collected  here  for  reference. 
Notice  that  the  second  and  fifth  are  special  cases  of  the 
first,  and  the  first  formula,  for  n  equal  to  a  whole  number 
only,  is  a  special  case  of  the  fourth, 

(1  +  5)»  =  1  +  nd, 
+  5)  =  1  -  5, 


1/1  +  d  =  1  + 


tan  5  _  sin  3  _ 

~<T      ~T 

75.  Questions  and  Exercises.  —  1.  Find  the  value  of 
(Suggestion:  divide  both  numerator  and  denom- 
inator by  800  in  order  to  convert  the  latter  into  1  —  5.) 

2.  Find  the  value  of  .504  X  .498.  (Suggestion:  the 
first  of  these  is  a  little  more  than  1/2,  i.  e.,  is  equal  to 
1/2  multiplied  by  a  little  more  than  one;  likewise,  the 
second  is  a  little  less  than  1/2.) 


VI  SMALL  MAGNITUDES  89 

3.  Write  the  general  formula  for  the  value  of  the  ex- 
pression (1  +  5i)"(l  +  52)n(l  +  53)p(l  +  54)9  •  •  ••    How 
many  of  the  formulae   that    are  given  in  §  74   does  it 
include? 

4.  When  warmed  up  to  ordinary  room  temperature 
the  brass  scale  of  a  barometer  is  too  long;  as  a  result  its 
reading  falls  short  of  the  correct  number  of  centimetres 
and  needs  to  be  multiplied  by  1.00037.     The  mercury, 
however,  expands  at  a  different  rate  from  the  brass  and 
tends  to  make  the  reading  too  great;  to  correct  this 
error  alone  the  reading  must  be  divided    by  1.00364. 
The    correct    height    of   the    mercury   will    accordingly 
be  obtained   if   the   observed   height  is   multiplied   by 
1.00037/1.00364.     What  is  the  percentage  of  difference 
between  the  corrected  reading  and  the  original  reading? 
(Suggestion:  call  the  observed  height  unity.) 

5.  In  the  equations  of  Fig.  14  eliminate  x  instead  of  d 


and  show  that  1  +  d  =  Vm^/mi.  Then  suppose  that 
m2  =  Wi  +  A  and  show  that  1  +  5  =  1+  A/2w,  and 
hence  that  25  =  A/m.  If  the  two  weighings  of  your 
avoirdupois  pound  (§71)  differed  by  0.3  percent  how 
much  do  the  two  arms  of  the  balance  differ  in  length? 

6.  If  V  10  =  3.16  find  the  value  of  7r2.     (Suggestion: 
v  =  3.14  =  3.16(1  -  2/316)). 


VII.     THE   SLIDE   RULE 

Apparatus. — A  slide  rule  provided  with  A,  B,  C,  D, 
L,  S,  and  T  scales,  a  runner,  and  a  list  of  equivalent 
measures. 

76.  Addition  by  Means  of  Two  Scales. — If  two  scales 
of  centimetres  should  be  laid  parallel  so  that  the  zero  of 
the  second  one  coincided  with  the  seventh  division  of  the 
first  (Fig.  16)  it  would  be  evident  that  the  fifth  division 


12345678 

i        i        i        i        i        i               i 

9 

to 

ft 

12 

13 

1 

i 

2 

i 

3 

i 

4 

i 

5 

i 

6 

i 

7 

i 

FIG.  16.  ADDITION  WITH  UNIFORM  SCALES. — By  starting  afresh 
at  7  of  the  original  scale  and  measuring  5  (or  any  other  number,  n) 
on  the  new  scale  we  reach  the  distance  of  7  +  5  (or  7  +  n,  as  the 
case  may  be)  on  the  original  scale.  It  should  be  noticed  that  this 
illustrates  subtraction  as  well  as  addition:  12  —  5  =  7,  also  the 
difference  between  any  other  pair  of  opposite  scale  numbers  is  equal 
to  7. 

of  the  second  would  be  opposite  the  twelfth  division  of 
the  first.  In  other  words,  when  a  length  of  five  has  been 
added  in  this  manner  to  a  length  of  seven  the  result  can 
be  seen  at  a  glance  to  be  equal  to  twelve.  Notice  also 
that  the  arithmetical  difference  between  scale  divisions 
that  lie  opposite  each  other  is  everywhere  the  same,  and 
that  it  is  equal  to  the  number  on  the  first  scale  which  is 
opposite  the  beginning  of  the  second. 

77.  Multiplication  with  Logarithmic  Scales. — If  the 
same  experiment  is  tried  with  two  scales  whose  divisions 
are  not  a  succession  of  whole  numbers  but  are  the 
logarithms  of  such  a  series  the  result  will  be  different, 

90 


VII  THE  SLIDE  RULE  91 

for  adding  logarithms  corresponds  to  multiplying  natural 
numbers.     Accordingly  if  we  start  at  log  7  (Fig.  17)  and 


t5    2        3     4    5  6  7  8  9  K)        15    20      30   40      60    80  100      150 

•  •"- !  •  !  •  '''-''KVl^Mi'M.  ^'.i.'iti'.iVI'.'il'V/.'.i'-v 

I      1.5    2       3456  78910      15    20  25 


FIG.  17.  MULTIPLICATION  WITH  LOGABITHMIC  SCALES. — The 
distances  on  each  scale  which  are  marked  1,  2,  3,  etc.,  are  really  equal 
to  log  1,  log  2,  log  3,  etc.  Since  addition  of  logarithms  effects  multi- 
plication of  their  natural  numbers  it  will  be  seen  that  the  number 
above  the  5  of  the  lower  scale  is  not  7  +  5  as  in  Fig.  16,  but  is  equal 
to  7  X  5,  or  35.  Since  subtraction  of  logarithms  corresponds  to 
division  it  will  be  evident  that  the  opposite  pairs  of  numbers  have 
their  quotients  equal  to  seven,  instead  of  their  differences. 

measure  a  further  distance  of  log  5  we  shall  come  out  with 
log  7  +  log  5,  which  is  not  equal  to  log  (7  +'  5)  but  to 
log  (7X5).  Notice  not  only  that  5  on  the  second  scale 
comes  opposite  35  on  the  first,  but  also  that  quotients 
of  corresponding  numbers  are  everywhere  equal  to  7, 
just  as  differences  are  in  Fig.  16;  and  furthermore,  the 
upper  scale  in  Fig.  17  forms  a  multiplication  table  (a 
seven  table  in  this  case)  for  the  numbers  on  the  lower 
scale  just  as  it  did  an  addition  table  in  Fig.  16. 

78.  The  Slide  Rule. — The  apparatus  called  a  slide  rule 
is  essentially  a  ruler  containing  a  groove  in  which  is  a 
movable  slide.  Logarithmic  scales  are  so  marked  that 
one  of  them  (the  "A  scale")  can  be  held  stationary 
while  another  (the  "  B  scale  ")  can  be  placed  in  any 
required  position  below  it  (Fig.  18). 

Two  scales  (C  and  D)  along  the  lower  edge  of  the  slide 
can  be  used  in  the  same  way  and  can  be  read  a  little  more 
accurately  on  account  of  their  subdivisions  being  larger, 
but  the  two  upper  scales  (A  and  B)  will  be  found  the  most 
convenient  for  general  use.  Each  one  of  them  is  really 


92 


THEORY  OF  MEASUREMENTS 


§78 


f|e 


CQO 


•t 


II 


"->      M      QJ 

SB  §> 


a  B.-S 

«3    ft  _. 

a 


-a 


f-Jlil 

!yl!; 

^  Q    O     «    3 
I     !i   GO    *3    A-  ' 


oq 


s 


^  T3 

gl  « 

H  «  ^  1  ^  -3 

_  •  •    +2     ^5     c-\ 


OS  -^ 

•i"^ 
ft  2  g 

o  cti  .2 


.S 


VII  THE  SLIDE  RULE  93 

two  complete  logarithmic  scales,  as  will  readily  be  seen, 
but  its  right-hand  half  is  often  marked  10  to  100  instead 
of  1  to  10.  In  calculating  with  the  aid  of  the  slide  rule 
it  is  advisable  to  ignore  all  decimal  points  while  using  the 
apparatus,  and  to  point  off  the  required  answer  by  making 
a  rough  preliminary  calculation  mentally,  as  illustrated 
in  §  14.  It  is  then  possible  to  use  either  the  7,  for  ex- 
ample, in  the  left-hand  part  of  the  A  scale,  or  the  70  (or 
7)  in  the  right-hand  part,  whichever  may  happen  to  be 
the  more  convenient,  in  order  to  represent  7000,  or  700, 
or  .07,  or  any  other  number  whose  significant  figure  is  7. 

By  consulting  Fig.  17  find  the  value  of  70  X  20.  Ans. : 
1400.  Find  3|  -f-  5.  Ans.:  0.7,  because  it  is  evident 
that  the  quotient  must  be  somewhat  less  than  1.  Find 
the  approximate  value  of  .0007  X  1.4.  Ans.:  nearly 
.0010.  Find  70  X  45.  Ans.:  as  nearly  as  the  value  can 
be  read  from  the  diagram,  it  seems  to  be  32;  since  70X40 
=  2800  the  required  product  must  be  3200,  the  ciphers 
not  being  significant.  Two-figure  accuracy  is  the  highest 
that  can  be  obtained  from  Fig.  17. 

79.  Reading  a  Logarithmic  Scale. — The  ordinary  slide 
rule  is  usually  constructed  so  carefully  that  three-figure 
accuracy  is  always  attainable,  although  it  requires  the 
"  estimation  "  (§  46)  of  halves  or  fifths  or  even  tenths 
of  the  graduated  intervals.  This  is  sufficient  for  the 
great  majority  of  practical  calculations  in  every-day  life; 
if  four-figure  accuracy  or  greater  is  required  it  is  usually 
more  convenient  to  use  logarithm  tables,  but  a  slide-rule 
makes  a  3-figure  logarithm  table  superfluous  because  it 
can  be  used  with  greater  rapidity. 

It  will  be  observed,  both  in  Fig.  18  and  on  the  slide  rule 
itself,  that  the  distances  from  log  1  to  log  2,  log  2  to  log  3, 
log  3  to  log  4,  etc.,  grow  progressively  smaller  and 
smaller,  so  that  those  near  the  left-hand  end  of  the  scale 


94  THEORY  OF  MEASUREMENTS  §79 

can  be  more  finely  subdivided  than  those  toward  the 
right-hand  end.  Thus  it  should  be  noticed  that  on  the 
A  or  B  scale  of  the  ordinary  slide  rule  the  third  subdivision 
to  the  right  of  7  or  70  is  73,  while  on  the  C  or  D  scale  it 
is  715;  but  the  third  subdivision  to  the  right  of  1  rep- 
resents 106  on  the  upper  scales  and  103  on  the  lower 
ones.  To  avoid  mistakes  in  picking  out  the  position  of 
a  given  number  it  is  best  to  find  both  the  next  larger 
single  figure  and  the  next  smaller  one;  then  look  for  the 
long  .5  graduation  which  is  about  half-way  between 
them. 

Find  365  on  the  A  scale  of  your  slide  rule  by  locating 
first  3,  then  4,  then  3.5,  then  3.6  and  3.7,  and  finally  365. 
Find  235  in  the  same  way;  then  by  estimation  set  the 
transparent  runner  with  its  vertical  line  as  nearly  as 
possible  at  238.  Move  the  slide  so  that  the  end  (1) 
of  the  B  scale  comes  under  this  238,  and  see  if  210  comes 
just  under  500.  Set  the  transparent  runner  at  128; 
then  at  106,  and  109;  Repeat  the  work  of  this  para- 
graph using  the  D  and  C  scales  respectively  instead  of  the 
A  and  B  scales. 

In  reading  a  given  position  on  the  scale  it  is  likewise 
best  to  read  the  next  larger  single  figure  and  the  next 
smaller  one,  also  the  half-way  point  between  them. 

Set  the  left-hand  1  of  the  B  scale  at  random  somewhere 
between  1  and  2  of  the  A  scale.  Then  read  its  position 
accurately;  also  the  positions  of  2,  3,  4,  5,  6,  7,  8,  and  9. 
On  some  slide  rules  the  subdivisions  between  1  and  2  of 
the  C  and  D  scales  are  marked  1,  2,  3,  •  •  •  instead  of 
1.1,  1.2,  1.3,  etc.  If  this  is  the  case  with  your  slide  rule 
it  will  be  necessary,  when  looking  for  a  number  such  as  7, 
to  see  that  it  belongs  in  the  series  that  runs  7,  8,  9,  10, 
out  to  the  right-hand  end  of  the  rule,  and  not  in  the 
series  that  runs  merely  from  1.0  to  2.0. 


VII  THE  SLIDE  RULE     -  95 

80.  Multiplication. — To    find    the    product    of    two 
numbers  with  the  slide  rule  either  end  of  the  B  scale  should 
be  set  opposite  one  factor  on  the  A  scale,  then  the  other 
factor  on  the  B  scale  will  be  opposite  the  required  product 
on  the  A  scale.     Try  this  process  with  small  numbers  by 
setting  1  on  B  opposite  3  on  A  and  observing  that  the 
position  of  2  on  B  gives  the  product  6  on  A. 

Multiply  2  X  4;  2  X  5;  2  X  6;  3  X  9;  7  X  8.  Find 
7  X  13,  and  7  X  16,  remembering  that  the  two  halves  of 
the  B  scale  (and  also  those  of  the  A  scale)  are  identical, 
so  that  if  16  (or  1.6)  when  taken  on  the  right  half  of  the 
B  scale  falls  beyond  the  end  of  the  A  scale  the  same 
result  can  be  read  over  16  (or  1.6)  on  the  left  half. 

Multiply  1.5  by  2;  1.8  X  2;  1.7  X  2.1  (estimate  the 
third  figure  of  the  product);  17.8  by  2;  1.79  by  2.53. 

Find  0.6  X  183 .(caution:  notl.6X  183;  see  end  of  §  79) ; 
7.3  X  1.09;  325.  X  106.5;  0.073  X  0.0016.  If  you  have 
any  difficulty  in  pointing  off  the  last  product  master  it 
before  going  any  further.  The  use  of  "  standard  form  " 
often  makes  the  preliminary  checking  easier;  thus  in  the 
last  two  examples  given  3  X  102  multiplied  by  1  X  102 
gives  3  X  104  or  3000;  and  (7  +)  X  10~2  multiplied  by 
(2  -)  X  10-3  gives  14  X  10~5,  or  .00014. 

81.  Division. — In  Fig.  16,  above,  notice  that  a  length 
of  twelve  may  be  considered  as  being  laid  off  from  left  to 
right,  and  then,  beginning  at  the  point  12,  a  length  of  5 
is  laid  off  to  the  left,  or  subtracted  from  the  original  12, 
giving  a  final  result  of  7.     Similarly,  in  fig.  17,  log  35 
minus  log  5  equals  log  7.     The  rule  for  division  is  ac- 
cordingly: place  the  divisor  on  B  under  the  dividend  on 
A  and  read  the  answer  on  A  over  either  end  (or  the 
middle)  of  the  B  scale.     Such  directions  should  never  be 
memorized  by  the  student,  but  he  should  practice  the 
process  until  he  becomes  familiar  with  it,  and  in  case  of 


96  THEORY  OF  MEASUREMENTS  §82 

doubt  should  experiment  with  small  numbers  where  the 
answer  is  obvious  beforehand. 

Divide  35  by  5;  30  by  5;  30  by  4;  11  by  4;  11  by  7; 
11  by  10;  11  by  11;  11  by  12;  11.8  by  99.;  114.  by  3.4; 
3.4  by  114. 

82.  Ratio  and  Proportion. — In  Fig.  16  it  will  be  seen 
that  8  —  1  =  9  —  2  =  10  —  3  =  7.  Since  subtraction 
of  logarithms  corresponds  to  division  of  natural  numbers 
Fig.  17  shows  that  14  -h  2  =  21/3  =  28  :  4  =  35/5, 
etc.  That  is,  with  the  slide  set  in  a  given  position  any  two 
opposite  numbers  are  in  the  same  ratio  as  any  other  two. 
Test  this  on  the  slide  rule  with  small  numbers;  for  example, 
set  6  under  2  and  notice  that  15  is  under  5,  for  6  : 2  : :  15  : 5. 
Solve  the  following  proportions  by  setting  the  rule  so 
that  the  answer  is  always  found  on  the  A  scale.* 


6  :2  : 

2:6: 

3  :2  ::9  :x  : 


15  :x, 
15  :  x, 
12  :y  ::  10  :  z. 


Solve  31  :  750  : :  .005:  x\  first  notice  that  750  is  about 
twenty  times  as  large  as  31.  Solve  2300  :  .036  ::  990  :  x. 
If  26  inches  =  66  centimetres  solve  the  following  equa- 
tion as  accurately  as  possible :  26  :  66  : :  1  :  x.  What  is 
the  length  in  centimetres  of  1  inch?  From  the  slide  rule 
find  the  approximate  length  of  4  inches.  How  many 
centimetres  in  41  inches? 

Set  1  precisely  under  the  special  mark  that  indicates 
TT  (=  3.142)  and  notice  that  7  is  nearly  under  22  because 
22/7  is  approximately  equal  to  TT.  Look  along  the  scales 
for  another  ratio  which  is  equal  to  TT,  and  find  one  that 
is  not  numerically  equal  to  22/7  or  to  3.142857.... 

*  To  avoid  uncertainty  about  which  scale  should  be  read  it  is 
advisable  to  get  into  the  habit  of  setting  the  slide  rule  so  that  the 
answer  will  always  be  found  on  one  of  the  fixed  scales,  -1  or  D;  not 
on  either  of  the  movable  ones,  B  or  C. 


VII  THE  SLIDE  RULE  97 

Then  find  its  decimal  value  by  unabridged  division  carry- 
ing out  the  work  until  the  figures  become  different  from 
those  of  the  correct  value  of  TT,  as  was  done  in  §  49. 

83.  Equivalent  Measures. — Turn  to  the  back  of  the 
slide  rule  and  find  a  statement  about  centimetres  and 
inches.  It  will  usually  be  given  in  tabular  form,  such  as 
cm:  in.  ...  26:  66,  together  with  various  other  data; 
if  not,  use  the  table  given  here.  Find  a  statement  of  the 

26  in  =  66  cm  30  atmo  =  31k/cm2 

292  ft  =  89  m  128  lb/in2  =  9  k/cm2 

35  yd  =  32  m  500  lb/in2  =  34  atmo 
87  mi  =  140  km 

340  ft-lb  -  47  kgm-m 

31  in2  =  200  cm2  134  hph  =  100  kwh 

140  ft2  =  13  m2  67  kwh    =  58000  Cal 

990  HL  =  61  cm3  TT  =  355/113 

23/3  =  680cm3  V  TT  =  39/22 
36  in3  =  590  cm3 

14  gal  =  53000  cm3  Zl  -  57°  11'  45" 

-  3437'.7 
108  gr  =  7  gm  =  206264//,8 

9  %  =  280  gm  1°  =  1.74533  X  10~2 

1940  av  oz  =  55  kgm  1'  =  2.90888  X  10~4 

22  Ib  =  10  kgm  I"  =  4.84814  X  10~6 

TABLE  OF  EQUIVALENTS  FOR  USE  WITH  THE  SLIDE  RULE. — The 
particular  numbers  are  chosen  so  as  to  be  accurate  as  far  as  three 
significant  figures,  at  least.  Thus  not  only  does  26  in.  equal  66  cm., 
but  also  26.0  in.  =  66.0  cm.  The  value  of  TT  is  not  only  355/133, 
but  also  3550/1330,  and  35500/13300,  even  to  355000/133000,  as 
may  easily  be  verified  by  the  process  of  abridged  division. 

relation  between  pounds  and  kilograms  and  calculate 
your  own  weight  in  kilograms,  remembering  to  set  the 
slide  rule  so  that  the  answer  will  be  found  on  the  A 
scale. 

84.  Reciprocals. — Set    4    or    3    or    any    other    small 


98  THEORY  OF  MEASUREMENTS  §86 

number  on  B  under  1  on  A,  and  notice  that  above  1  (or 
10  or  100)  of  the  B  scale  will  be  found  .250  or  .333  or  the 
reciprocal  of  whatever  small  number  was  used.  Verify 
1  -j-  7  =  .142857,  as  accurately  as  the  apparatus  .will 
allow. 

85.  C  and  D  Scales. — Experiment  with  any  small  num- 
ber, using  the  C  and  D  scales,  and  show  where  it  must  be 
set  in  order  to  find  its  reciprocal  on  the  (fixed)  D  scale. 
Notice  that  if  8  on  C  is  set  over  10  on  D  the  value  of  1/8 
will  be  found  under  1,  but  if  2  is  set  over  1  the  value  of 
1/2  will  be  found  under  the  10  instead. 

Try  multiplication,  division,  proportion,  and  equiva- 
lents on  the  C  and  D  scales,  remembering  to  set  the  slide 
so  that  the  answer  always  comes  on  the  stationary  (D) 
scale.  If  the  method  for  any  of  these  processes  has  been 
forgotten  experiment  with  small  numbers  so  that  you 
know  the  required  answer  beforehand.  In  multiplica- 
tion if  the  answer  runs  off  the  end  of  the  rule  set  10 
instead  of  1  on  C  opposite  the  first  factor,  and  in  division 
read  the  result  under  10  instead  of  under  1  when  neces- 
sary. If  the  fourth  term  of  a  proportion  cannot  be  read 
the  runner  must  be  placed  over  the  end  of  the  C  scale  and 
the  slide  then  moved  so  that  the  other  end  of  the  C  scale  is 
under  the  runner.  As  an  example,  set  26  on  C  over  66  on 
D  and  show  that  the  number  of  centimetres  in  5  inches 
is  12.7. 

86.  Squares  and  Square  Roots. — Set  the  slide  so  that 
the  ends  of  the  B  and  C  scales  coincide  with  the  ends  of 
the  A  and  D  scales.     Then  move  the  runner  so  that  its 
vertical  line  falls  on  9  of  the  C  and  D  scales  and  notice 
that  it  also  comes  on  92  or  81  of  the  A  and  B  scales.     Set 
it  at  8,  7,  6,  5,  etc.,  on  the  lower  scales  and  notice  that 
the  number  on  the  upper  scales  just  over  n  on  the  lower 
ones  is  always  n2.     For  a  slide  rule  that  has  no  runner  set 


VII  THE  SLIDE  RULE  99 

1  on  C  over  n  on  D  and  1  on  B  will  indicate  n2  on  A. 
Read  the  square  of  12;  of  13;  of  19.  Find  the  (four- 
figure)  square  of  43,  knowing  that  the  last  figure  must 
be  9. 

Under  any  number  on  the  A  (or  B)  scale  will  be  found 
its  square  root  on  the  D  (or  C)  scale,  but  it  must  be  re- 
membered that  any  given  arrangement  of  significant 
figures  has  two  different  square  roots,  according  to  how 
it  is  pointed  off  (compare  §  13,  ex.  41).  For  example, 
V  1500  =  40-;  V  150  =  12+;  V  15  =  4  - ;  V  1.5 
=  1.2  +;  V-15  =  .4  -;  V  .015  =  .12  +;  etc.  Notice 
that  under  150  of  the  left-hand  half  of  the  A  scale  one 
of  these  roots,  122,  is  found;  while  the  other  root,  387, 
occurs  under  the  150  of  the  right-hand  half.  Since  one 
of  these  is  about  three  times  as  large  as  the  other  the 
simple  precaution  of  making  mentally  a  rough  pre- 
liminary calculation  of  the  root  will  avoid  the  possibility 
of  obtaining  the  wrong  number.  For  example,  is  the 
square  root  of  .036  given  by  the  significant  figures  19  or 
60,  and  how  should  they  be  pointed  off? 

87.  Compound  Operations. — A  quotient  such  as  m/n 
is  found  by  setting  n  on  the  B  scale  under  m  on  the  A 
scale  and  reading  the  value  opposite  the  end  of  B.  (If 
this  is  not  perfectly  obvious  try  it  with  small  numbers.) 
If  m/n  is  further  to  be  multiplied  by  some  other  number, 
say  Xj  notice  that  the  slide  rule  does  not  need  to  be  re-set 
as  it  is  already  arranged  so  that  the  required  product 
will  be  found  over  x  of  the  B  scale.  Find  two  sevenths 
of  thirteen  in  this  way  and  notice  particularly  that  it  is 
not  necessary  to  read  the  value  of  the  fraction  2/7; 
merely  set  7  under  2  and  find  the  required  answer  over  13. 

Similarly,  m?n  will  be  found  on  A  opposite  n  of  the 
B  scale  if  1  on  C  is  set  at  m  on  D;  and  Vmn  will  be  found 
on  D  under  n  on  B  if  1  on  B  is  set  to  m  on  A. 


100  THEORY  OF  MEASUREMENTS  §88 

A  series  of  fractions  like  a/m,  b/m,  c/m,  d/m,  -  •  • ,  can 
be  read  off  merely  by  setting  the  slide  rule  for  1/w  and 
looking  opposite  a,  b,  c,  d, 

The  slide  rule  is  usually  made  with  two  "  cylinder 
points  "  marked  on  the  C  scale  at  i/4/ir  and  i/40/Tr. 
By  placing  either  one  of  these  opposite  the  diameter  of  a 
cylinder  the  length  of  the  cylinder  on  B  will  be  found  to 
indicate  its  volume  on  A. 

There  is  usually  a  special  mark  for  TT  on  the  left-hand 
half  of  the  A  and  B  scales,  and  for  ?r/4  (or  .7854)  on  the 
right-hand  half.  By  placing  the  end  of  the  B  scale 
opposite  the  latter  the  area  of  any  circle  will  be  found 
on  A  opposite  the  diameter  on  D. 

88.  Determination  of  Circular  Functions. — In  most 
slide  rules  the  back  of  the  sliding  part  is  provided  with 
three  scales,  which  are  named,  and  sometimes  marked, 
8,  L,  and  T,  from  above  downward.  If  the  slide  is 
placed  so  that  the  ends  of  the  S  and  T  scales  coincide  with 
the  ends  of  the  A  and  D  scales  respectively,  the  sine  of 
any  angle  from  1°  to  90°  will  be  found  on  A  opposite  the 
number  of  degree  and  minutes  on  S;  and  the  tangent  of 
any  angle  from  6°  to  45°  will  be  found  on  D  opposite  the 
number  of  degrees  and  minutes  on  T.  The  decimal  point 
is  located  by  recalling  the  facts  that  sin  90°  =  1,  tan 
45°  =  1,  and  if  two  angles  differ  only  slightly  their 
sines  (or  tangents)  will  also  be  only  slightly  different. 

By  using  the  slide  rule  show  that  sin  70°  =  .940;  write 
the  values  of  sin  50°;  sin  30°;  tan  30°;  sin  11°  30';  tan 
11°  30';  sin  6°;  sin  5°;  sin  1°. 

Since  tan  (45°  +  a)  and  tan  (45°  -  a)  are  reciprocals 
of  each  other  (prove  by  substituting  45  +  a  for  x  in  the 
equation  of  §  45;  12,  c),  tangents  of  angles  greater  than 
45°  are  easily  obtained  from  the  slide  rule.  For  example, 
to  find  tan  49°,  which  is  tan  (45°  +  4°),  set  41°,  which  is 


VII  THE  SLIDE  RULE  101 

45°  —  4°,  opposite  10  on  D  and  read  the  required  value, 
1.15,  on  D  opposite  the  left-hand  of  the  T  scale. 

For  sines  less  than  .01  and  tangents  less  than  0.1  differ- 
ent types  of  slide  rule  employ  different  methods,  usually 
based  on  the  formulae  sin  <5  =  tan  d  =  d  (§  72).  If  no 
special  marks  for  angles  are  found  on  the  C  scale  or  the  S 
scale  use  the  equations  for  1°,  1',  and  I"  in  §83. 

89.  Determination  of  Logarithms  and  Antilogarithms. 
— Set  1  on  C  opposite  any  number,  n,  on  D,  and  log  n 
will  be  found  on  L  opposite  a  special  mark  on  the  back 
of  the  slide  rule.     Try  this  with  small  numbers  whose 
logarithms    are    already   known;    e.    g.,    log   3  =  0.477. 

90.  Questions  and   Exercises. — 1.  If  decimal  points 
are  disregarded  one  square  root  of  a  given  arrangement  of 
significant  figures  is  stated  (§  86)  to  be  about  three  times 
as  large  as  the  other.     What  is  the  exact  value?     Why? 

2.  Prove  that  the  volume  of  a  cylinder,  irr2l,  is  cor- 
rectly obtained  by  the  process  given  in  §  87. 

3.  Explain  how  it  is  that  the  process  in  §  88  for  finding 
tan  49°  really  gives  the  reciprocal  of  tan  41. 

4.  Set  1  on  the  C  scale  over  a  tentative  cube  root  of  n 
and  see  whether  %  on  B  comes  under  n  on  A.     Practice 
this  method  of  finding  cube  roots.     In  what  way  would 
the  marks  ^10  and  ^100  in  Fig.  18  be  helpful? 

5.  Find  a  way  to  use  the  marks  "  in/100,  40  20  0" 
opposite  32  ±  of  the  A  scale  in  Fig.  18  in  order  to  reduce 
"  American  Wire  Gauge  "  to  actual  diameters  of  the  wires 
in  hundredthsof  an  inch.     Start  from  the  following  data: 

No.  0  =  .325  inch,  no.  1  =  .289,  no.  2  =  .258,  no. 
3  =  .229,  no.  4  =  .204;  no.  8  =  .128,  no.  12  =  .081,  no. 
16  =  .051,  no.  20  =  .032,  no.  24  =  .020,  no.  28  =  .013, 
no.  30  =  .010.  Notice  the  marks  "cm/100"  that  are 
located  [log]  25.4  times  as  far  to  the  right,  and  show  that 
the  diameter  of  no.  18  wire  is  very  nearly  1.00  mm. 


VIII.     GRAPHIC   REPRESENTATION 

Apparatus. — A  pencil  with  a  sharp  point  for  marking 
positions  accurately  on  the  squared  paper  of  the  notebook. 

91.  Indication   of  a  Point  by  Two    Numbers. — The 
position  of  any  point  on  the  surface  of  the  earth  may  be 
located  by  two  numbers.     One  of  these,  the  longitude 
of  the  point,  expresses  its  distance  to  the  east  or  west  of 
an  arbitrary  line,  the  meridian  of  reference.     The  other, 
its  latitude,  gives  its  distance  north  or  south  from  a 
definite  base  line,  the  equator. 

92.  Representation  of  Two  Numbers  by  a  Point. — In 
almost  all  branches  of  science  a  process  which  is  just  the 
opposite  of  that  given  above  has  been  found  to  be  of  very 
great  value:   instead  of  using  two  numbers  to  describe 
the  location  of  a  point  the  method  is  reversed  and  any 
two  related  numbers  are  represented  diagrammatically 
by  the  position  of  a  point.     For  example,  in  order  to 
indicate  that  the  out-door  temperature  on  January  eighth 
was  21°  F.,  a  point  could  be  marked  on  a  sheet  of  paper 
at  a  distance  of  8  arbitrary  units  from  the  left-hand  edge 
of  the  paper  and  at  the  same  time  21  units  above  the 
bottom  of  the  sheet.     If  the  temperature  had  fallen  to 
17  on  Jan.  9,  another  point  could  be  marked,  located  9 
units  from  the  left-hand  edge  and  17  units  above  the 
lower  edge  of  the  paper. 

93.  Representation  of  Two  Variables  by  a  Line. — By 
continuing  the  process  of  marking  down,  or  "  plotting  " 
a  new  point  for  each  successive  day  and  its  temperature 
a  series  of  points  would  be  obtained;  and  if  the  tempera- 
tures should  be  observed  at  more  frequent  intervals, 
every  hour  or  every  minute,  the  points  would  come  so 

102 


VIII 


GRAPHIC  REPRESENTATION 


103 


close  together  that  they 
would  almost  make  a 
continuous  line.  The 
ups  and  downs  of  the 
irregular  line  would  in- 
dicate clearly  to  the  eye 
the  fluctuations  of  tem- 
perature that  correspond 
to  the  onward  march 
of  time,  which  would  be 
indicated  by  the  steady 
progress  of  the  line  from 
left  to  right.  The 
ob j ectionable  feature 
of  a  diagram  of  this 
kind  would  be  that  no 
temperature  below  zero 
could  be  represented. 


° 


ordinate. 


abscissa 
FIG.  20.  GRAPHIC  DIAGRAM. — 
The  two  numbers  6  and  4  are  repre- 
sented by  a  point  (P)  whose  coordi- 
nates are  the  abscissa,  or  z-value,  of 
6  units  measured  horizontally,  and 
the  ordinate,  or  ?/-value;  of  4  units 
measured  vertically.  Positive  values 
are  always  measured  to  the  right 
and  upward;  negative  ones,  to  the 
left  and  downward. 


94.  Graphic  Diagrams.— In  order  to  allow  the  repre- 
sentation of  negative  values  of  a  variable  such  as  tem- 
perature it  is  customary  not  to  measure  from  the  bottom 
of  the  paper  but  from  an  arbitrary  horizontal  line  ruled 
at  a  sufficient  height  to  allow  space  for  the  data  that  are 
to  be  indicated.  The  other  variable  is  not  always  time, 
but  may  be  a  changing  quantity  which  also  assumes 
negative  values,  so  that  a  vertical  line  of  reference  must 
be  ruled  at  some  distance  from  the  left-hand  edge  of  the 
paper.  A  graphic  diagram  consists  of  these  two  lines  of 
reference,  called  axes,  a  numerical  scale  along  each  of 
them,  and  the  point  or  assemblage  of  points  which  cor- 
responds to  the  numerical  values  that  are  to  be  repre- 
sented. The  vertical  distance  of  any  point  from  the 
horizontal  line,  or  x-axis,  is  called  the  ordinate  of  that 
point  (Fig.  20);  and  the  horizontal  distance  from  the 


104 


THEORY  OF  MEASUREMENTS 


vertical  line  of  reference,  or  y-axis,  to  this  ordinate  is 
called  the  abscissa  of  the  point  In  the  diagram  the 
point  P  has  an  abscissa  of  6  and  its  ordinate  is  4.  The 
abscissa  of  a  point,  being  measured  along  the  z-axis,  is 
sometimes  called  the  x-value  of  the  point;  and  the  or- 
dinate, or  vertical  distance,  is  called  the  y-value.  Con- 
sidered collectively,  the  two  distances  are  called  the 

coordinates  of  the  point. 
The  point  whose  coordi- 
nates are  both  zero,  viz., 
the  intersection  of  the 
two  axes,  is  called  the 
origin  of  the  graphic  dia- 
gram. 

95.  Practice  in  Plot- 
ting Points. — Figure  21 
shows  three  points  rep- 
resented by  small  dots 
and  three  others  marked 
by  crosses.  The  highest 
dot  has  an  z-value  of 


•X 


FIG.  21.  -POINTS  ON  A  GRAPHIC 
DIAGRAM. — The  points  shown  by 
small  dots  have  no  negative  values. 
The  points  shown  by  crosses  have 
negative  values  for  one  or  both  .of 
their  coordinates. 


+  5  and  a  y-value  of 
+  3;  when  it  is  consid- 
ered alone  it  may  be 
spoken  of  as  "  the  point 

(5,  3),"  the  coordinates  being  written,  x-value  first,  in 
parentheses  and  separated  by  a  comma.  Notice  that 
the  other  dots  are  located  at  (6,  0)  and  (7,  1).  One 
of  the  crosses  is  located  at  (—  2,  +3).  Write  down  the 
location  of  each  of  the  other  two. 

Draw  two  short  axes  in  your  notebook.  Mark  their 
positive  directions  X  and  Y,  as  in  Fig.  21.  Lay  off  a  short 
scale  of  positive  or  negative  numbers  along  each  axis, 
numbering  every  line,  or  every  other  line  or  every  fifth 


VIII  GRAPHIC  REPRESENTATION  105 

line,  as  you  prefer.  Make  a  dot  for  each  of  the  following 
points:  (1,  2);  (2,  1);  (3,  0);  (4,  -  1);  (5,  -  2).  Without 
drawing  new  axes  plot  the  following  points  with  small 
X 'son  the  same  diagram:  (4,0.5);  (0,  -  1.5);  (2,  -  0.5); 
(—  1,  —  2);  (1,  —  1);  (3,  0).  On  the  same  diagram  use 
small  +'s  for  the  following:  (-  0.5,  0);  (  +  0.5,  -  2); 
(-  2,  +  3);  (0,  -  1);  (-  1.5,  +  2);  (-  1,  +  1).  Frac- 
tional values  are  most  conveniently  plotted  by  drawing 
short  horizontal  and  vertical  lines  that  intersect  to 
make  a  +.  On  the  same  diagram  plot  the  following 
points  in  this  way  and  then  connect  them  by  a  smooth 
free-hand  curve: 

rr  =  0.6  1.0  2.0  2.3  3.0  3.7  4.0  5.0  6.0  7.0  7.7 
7/-3.0  3.1  3.1  3.0  2.5  2.0-1.6  1.0  0.5  0.6  1.0 

Take  one  of  the  following  tables,  I-V,  as  indicated  by 
your  instructor,  and  locate  all  of  its  points  by  means  of 
small  +'s  on  a  new  graphic  diagram.  Before  drawing 
the  axes  find  the  largest  and  (algebraically)  smallest  of 
the  re-values  in  the  table  and  see  that  the  ?/-axis  is  not 
drawn  too  far  to  the  right  or  left  to  leave  room  for  them. 
Examine  the  ^/-values  in  the  same  way  and  draw  the 
re-axis  as  closely  under  your  previous  notes  as  they  will 
allow.  Do  not  start  the  diagram  on  a  new  page  or  place 
it  so  as  to  use  up  an  unnecessary  amount  of  space.  Do 
not  draw  lines  to  connect  the  separate  points. 

96.  Orientation  of  a  Graphic  Curve. — On  the  squared 
paper  of  your  notebook  draw  a  rectangle  that  includes 
128  of  the  small  squares,  making  it  8  squares  wide  and 
16  squares  high.  Examine  the  table  on  the  next  page 
(Table  A)  and  locate  the  re-axis  and  i/-axis  so  that  none 
of  the  points  representing  the  tabular  values  shall  fall 
outside  of  the  limits  of  the  rectangle.  Draw  a  second 


106 


THEORY  OF  MEASUREMENTS 


§96 


X 

y         x 

y          ^ 

y          x 

y         x 

V 

0 

i          -  i 

+  1            0 

2                 1 

1                 1 

1 

-  1 

0                 1 

4                 1 

1         -  1 

0                2 

0 

0 

-  1       -2 

3                0 

-  1            2 

3            -  6 

0 

1 

0                 1 

5                 1 

-  2        -  2 

5            -  8 

-  2 

2 

1             -  1 

0            -3 

-  2        -  1 

3            -  6 

-  2 

3 

2            -3 

1             -2 

-  3            0 

1             -  8 

0 

4 

3            -  7 

0                0 

-  3            1 

0                2 

2 

3 

4            -6 

-  1            1 

-  4       -  1 

2                 0 

0 

2 

3            -2 

5                4 

-  1       -  2 

1             -3 

0 

2 

-  1       -  5 

2                 5 

-2       -  5 

0                0 

-  2 

4 

1             -  4 

1                 1 

-5       -4 

-  1       -3 

2 

5 

0                0 

5                 2 

-6       -  2 

-  1       -  7 

-3 

7 

0            -4 

-  1            2 

-  1            0 

-3       -  3 

3 

4 

-3       -5 

4                4 

-  3            1 

-  2       -  4 

-  1 

3 

-  4       -  4 

5            -  1 

0                3 

2            -  5 

1 

1 

-  7       -2 

-  1       -2 

-  1        -  3 

3            -  3  !  -  1 

5 

-  6       -  7 

-  1            0 

-6            1|-1        -2l-4 

2 

-3       -5 

3                 1 

-6       -32            -5-3 

3 

-  2       -3 

2             -  1 

-  4            1 

-3       -11 

4 

-  1       -  1 

3                 1 

-3            3 

3                 3-1 

2 

-  5       -  5 

-  1            2 

2            -  3    4            -  1  |  -"3 

6 

-  1       -3 

5                 1 

2            -  5     -  1        -  7  i  1 

6 

1             -  1 

2                 3 

-  4            03            -  4 

2 

5 

-  7       -  3 

-  1            2 

-  3       -  3     -  1        -2 

2 

6 

-  7       -  5 

1                  1 

0            -  1     -  1        -3|-5 

0 

-  7       -  1 

-  1            3 

0            -3 

1                 1 

-3 

1 

-6       -3 

3                 1 

-  1        -  1 

1             -  4 

-  4 

3 

-  7       -  5 

5 

1 

3            -  9 

-  1 

4 

-  5       -  1 

5 

-  3 

5            -  2 

-  1 

TABLE  I        TABLE  II       TABLE  III      TABLE  IV       TABLE  V 

rectangle  of  the  same  size  and  plot  the  values  of  Table  B 
in  it  without  going  beyond  its  boundaries.  Examine 
Tables  C  and  D  and  plot  a  graphic  diagram  for  each  one, 


X 

V 

0 

-5 

1 

-8 

2 

-  9 

3 

g 

4 

-  5 

5 

0 

6 

7 

-  1 

0 

-  2 

7 

TABLE  A 


X 

y 

£ 

y 

rr 

y 

0 

-  i 

0 

-  4 

0 

0 

1 

+  1 

1 

-5 

0.5 

0.25 

2 

3 

2 

-5.6 

1 

1 

3 

5 

3 

-6 

2 

4 

4 

7 

5 

-6.5 

3 

9 

5 

9 

-  1 

2 

4 

16 

-  1 

-  3 

-  1.5 

0 

5 

25 

-  2 

-  5 

-  2 

4 

-  1 

1 

-3 

-  7 

-  2 

4 

-  3 

9 

TABLE  B 

TABLE  C 

TABLE  D 

VIII 


GRAPHIC  REPRESENTATION 


107 


in  one  of  the  rectangles  and  with  the  axes  as  they  have 
already  been  drawn  if  possible,  otherwise  in  a  third  rec- 
tangle which  is  to  be  of  the  same  size  as  the  previous  ones 
but  in  which  the  scales  of  z-values  and  ^/-values  may  be 
condensed,  so  that  the  dimensions  of  each  square  may 
represent  two,  or  five,  or  ten  units,  as  may  be  most 
convenient,  or  may  be  expanded,  so  that  one  unit  may  be 
represented  by  two  or  more  times  the  length  of  a  single 
square. 

In  the  next  table  notice  that  the  ^-values  lie  between 
the  extreme  values  of  +  42  and  +  48.  In  such  cases 
there  is  usually  no  objection  to  dispensing  with  the  a>axis 


FIG.  22.  GRAPHIC  DIAGRAM  WITH  CONDENSED  SCALES. — Notice 
that  the  axvalues  have  been  so  condensed  that  the  curve  does  not 
extend  far  to  the  right ;  and  the  7/-values  have  been  condensed  to  the 
same  extent,  making  the  curve  relatively  flat. 

FIG.  23.  GRAPHIC  DIAGRAM.— Notice  how  both  scales  have  been 
arranged  so  that  the  same  table  as  was  illustrated  by  Fig.  22  now 
has  its  values  much  better  displayed. 


108  THEORY  OF  MEASUREMENTS  §97 

on  the  diagram  if  the  scales  are  plainly  indicated.  In 
Fig.  22  there  is  much  wasted  space  between  the  curve 
and  the  z-axis,  and  the  curve  itself  is  relatively  flat. 
Notice  how  both  of  these  objections  have  been  overcome 
in  Fig.  23. 

x  =  1  2  3  4  5  6  7  8  9  10  11  12  13  14 
y  =  42  42-*  43  431  43£  43£  43|  43|  44  44  f  45|  46|  47|  48 

TABLE  OF  VALUES  REPRESENTED  IN  FIGS.  22  AND  23. 

97.  Choice  of  Scales. — When  only  a  few  points  are  to 
be  plotted  and  the  range  of  extreme  values  is  not  very 
great,  as  in  tables  A,  B,  C,  and  D  of  §  96  it  is  advisable 
to  use  a  normal  scale  of  one  unit  for  each  square  of  the 
ruled  paper.  All  of  the  graphic  diagrams  in  this  chapter 
and  the  next  one  are  to  be  constructed  in  this  way  unless 
special  directions  are  given  to  the  contrary.  Condensed 
scales  are  most  useful  for  large  numerical  values,  includ- 
ing data  given  in  round  numbers,  and  for  keeping  a 
diagram  within  the  limits  of  a  definite  space.  Expanded 
scales  are  necessary  for  minute  values  and  those  which 
need  accurate  representation.  For  example,  when  the 
width  of  a  square  is  only  half  a  centimetre  (one  fifth  of 
an'  inch)  or  less  it  is  possible  to  divide  it  into  tenths  by 
estimation  and  to  show  a  perceptible  difference  in 
position  to  correspond  to  a  difference-  of  one  tenth  in 
numerical  values,  but  values  differing  only  by  a  number 
of  hundredths  or  thousandths  need  an  enlarged  scale  to 
make  their  variations  show  in  the  graphic  diagram.  In 
some  cases  the  importance  of  the  relation  or  ratio  of 
z-value  to  t/-value  makes  it  necessary  that  both  z-scale 
and  ?/-scale  shall  be  the  same,  whether  normal  or  con- 
densed or  extended.  In  other  cases  the  two  sets  of 
values  represent  measurements  which  are  mutually  in- 
commensurable (as  in  the  case  of  time  and  temperature, 


VIII  GRAPHIC  REPRESENTATION  109 

§§92  and  93),  so  that  it  is  theoretically  of  no  importance 
whether  the  two  scales  correspond  or  not.  Where  the 
slant  of  the  curve,  or  of  some  important  part  of  it,  is 
fairly  uniform,  however,  it  is  often  more  satisfactory  to 
choose  the  scales  in  such  a  way  as  to  make  the  slope 
approximately  45  degrees,  as  in  Fig.  23. 

98.  General  Principles  of  Plotting. — When  one  of  two 
variable  quantities  changes  as  the  result  of  changing  the 
other  it  is  customary  to  use  the  horizontal  scale  of  re- 
values for  the  independent  variable   and  the   vertical 
scale  of  ^/-values  for  the  dependent  or  resultant  variable. 
Thus,  for  a  graphic  diagram  of  temperature  and  time,  it 
is  natural  to  consider  the  temperature  as  being  the  result 
of  the  time  rather  than  the  time  as  depending  upon  the 
temperature.     Before  starting  to  draw  a  graphic  dia- 
gram the  largest  and  smallest  values  of  the  variables 
should  be  observed,  so  that  the  axes  can  be  located  in  a 
satisfactory  position  on  the  paper.     The  next  step  should 
invariably  be  to  lay  off  a  scale  along  each  axis  before 
plotting  a  single  point,  and  to  see  that  its  zero  comes  at 
the  "  origin  "  and  that  equal  numerical  intervals  are  always 
represented   by   equal   distances   on   any   one   scale,    the 
(positive)  z-values  always  increasing  from  left  to  right, 
and  the  ^/-values  from  below  upward.     This  is  especially 
important  when  plotting  a  curve  from  a  table  like  No.  9 
on  page   114,   where  the  ^-values   are   given   at   larger 
intervals  in  one  part  of. the  table  than  in  another  part. 

99.  Representation    of    Tabular    Values. — After    the 
points  corresponding  to  a  set  of  tabular  values  have  been 
located  on  a  graphic  diagram  it  is  customary  to  draw  a 
straight  line  from  each  point  to  the  adjacent  point  on  the 
left  or  right,  making  a  broken  line  for  the  "  curve  "  that 
shows  the  fluctuations  in  the  value  of  the  dependent 
variable.     If  all  the  points  lie  on  a  smooth  curve  it  is 


110 


THEORY  OF  MEASUREMENTS 


§100 


hour 

temperature 

A.M. 

P.M. 

12 
1 
2 
3 
4 
5 
6 

36.9°  C. 
36.8 
36.8 
36.6 
36.4 
36.5 
36.6 

37.4 
37.4 
37.6 
37.5 
37.5 
37.6 
37.6 

7 
8 
9 
10 
11 
12 

36.8 
36.9 
37.1 
37.2 
37.2 
37.4 

37.7 
37.6 
37.4 
37.4 
37.2 
36.9 

advisable  to  draw  such  a  curve  as  evenly  as  possible,  but 
the  experimental  values  given  in  a  table  are  apt  to  show 

little  irregularities  which  make 
it  impossible  to  draw  a  smoothly 
flowing  curve  through  their 
graphic  points.  In  such  cases 
the  broken  line  serves  the  pur- 
pose of  visually  assembling  the 
points  in  a  series,  but  is  not  sup- 
posed to  indicate  that  interme- 
diate points,  if  obtainable,  would 
lie  exactly  along  the  straight 
lines. 

Draw  a  graphic  diagram  in 
which  the  horizontal  scale  rep- 
resenits  the  24  hours  of  a  single 
day,  beginning  at  midnight  and 
running  through  twelve  o'clock, 
noon,  to  the  next  midnight. 
For  the  vertical  ordinates  use 
the  temperatures  given  in  the 
table.  Connect  each  point  with 
the  next  by  means  of  a  straight 
line,  being  careful  not  to  omit  any  point.  Notice  how 
much  more  striking  and  "  graphic  "  the  diagram  is  than 
the  table;  how  it  shows  at  a  glance  facts  that  could  be 
gleaned  from  the  table  only  with  much  greater  effort. 

100.  Smoothing  of  a  Graphic  Curve. — In  the  temper- 
ature diagram  just  made  it  is  probable  that  the  little 
irregularities  of  temperature  are  due  to  accidental  causes 
and  would  not  be  exactly  repeated  in  taking  the  temper- 
ature of  another  individual  or  even  of  the  same  individual 
on  another  day.  In  such  cases  a  more  typical  picture  is 
given  by  drawing  a  smooth,  flowing  curve  in  such  a  way 


NORMAL  TEMPERATURE 
OF  THE  HUMAN  BODY. — 
Use  a  horizontal  distance 
of  one  square  to  represent 
one  hour,  and  a  vertical 
distance  of  one  square  to 
represent  one  tenth  of  a 
degree. 


VIII  GRAPHIC  REPRESENTATION  111 

that  it  passes  through  the  midst  of  the  scattering  points 
and  follows  their  general  upward  and  downward  trend 
without  necessarily  cutting  most  of  them  or  even  any 
one  of  them. 

It  ought  not  to  pass  below  most  of  the  points,  nor 
above  most  of  them,  but  should  leave  them  distributed, 
some  above  and  some  below,  as  evenly  as  possible,  sub- 
ject to  the  general  condition  that  it  must  be  a  smooth 
curve  that  does  not  show  even  a  tendency  to  resemble 
a  broken  line  either  by  indicating  the  irregularities  of 
the  table  in  the  form  of  wavelets  or  by  suggesting  an 
unduly  sharp  turn  or  "  corner  "  at  any  point. 

In  another  place  on  the  squared  paper  of  the  notebook 
plot  merely  the  points .  as  was  done  for  §  99 ;  then  draw 
a  smooth  curve  along  their  general  course,  outlining  it 
tentatively  with  a  light  pencil  mark,  erasing  and  cor- 
recting this  until  it  is  satisfactory,  and  then  tracing  it 
plainly  with  ink.*  It  should  show  not  more  than  one 
downward  loop  and  one  larger  upward  swing. 

This  process,  which  is  called  smoothing  a  graphic  dia- 
gram, is  advisable  only  when  one  can  be  reasonably 
certain  that  the  small  fluctuations  that  are  eliminated 
by  the  process  represent  unavoidable  experimental 
errors  or  chance  variations  or  else  that  they  are  due  to 
causes  which  are  negligible  in  the  case  that  is  under 
consideration.  Instances  have  occurred  in  which  even 
able  scientists  have  missed  the  discovery  of  important 
facts  on  account  of  the  "  smoothing  out  "  of  what  have 
seemed  to  be  only  accidental  irregularities. 

101.  Questions  and  Exercises. — 1.  What  shape  is  the 

*  Unless  special  drawing  apparatus  is  used,  a  curved  ink  line  can 
usually  be  drawn  more  neatly  if  it  is  made  dotted,  as  in  Fig.  54, 
instead  of  solid.  The  points  which  it  summarizes  may  be  marked 
rather  heavily  in  order  to  aid  the  eye,  but  this  must  be  done  uni- 
formly. 


112  THEORY  OF  MEASUREMENTS  §101 

curve  of  a  graphic  diagram  if  each  of  its  points  has  a 
2/-value  that  is  one  half  as  large  as  the  corresponding 
z-value? 

2.  What  is  the  most  probable  value  of  the  average 
temperature  of  a  healthy  individual  at  4  P.M.?     Would 
you  prefer  to  decide  the  matter  from  the  graphic  curve 
of  §  99  or  from  that  of  §  100? 

3.  Can  you  see  any  uniformity  or  law  in  the  arrange- 
ment of  the  points  obtained  from  any  of  the  Tables  A, 
B,  C,  and  D,  of  §  96?     Connect  them  with  smooth,  free- 
hand curves,  if  this  has  not  already  been  done.     The 
curve  of  Table  A  is  called  a  parabola,  B  is  a  straight  line, 
C  is  a  hyperbola,  and  D  is  a  parabola. 

4.  Select  any  two  points  on  the  graphic  diagram  of 
Table  B,  and  find  the  difference  in  their  x-values;  also 
the  difference  in  their  ^/-values.     Select  any  two  other 
points  on  the  same  line,  and  treat  them  in  the  same  way. 
What    relationship    always    exists    between    four    such 
differences? 

5.  How  great  is  the  gradient  (§  17)  of  the  line  repre- 
sented by  Table  B?     Write  an  expression  for  the  gradient 
of  any  straight  line  drawn  on  a  graphic  diagram  in  terms 
of  the  difference  in  z-values  and  the  difference  in  ^/-values 
of  any  two  points  that  are  located  on  the  line. 

6.  7,  8,  9,  10.     Plot  the  points  given  in  Tables  6,  7  (the 
numbers  that  are  enclosed  in  parentheses  may  be  omitted), 
8,  9,  and  10,  using  the  same  scale  for  the  z-axis  as  for 
the  2/-axis,    and  making  each    curve    large    enough  to 
cover  nearly  the  whole  page.     In  each  case  a  perfectly 
smooth  curve  can  be  drawn  which  will  pass  through  every 
point.     Number  6  is  called  a  sinusoid  or  sine  curve;  no. 
7  is  a  curve  of  tangents;  no.  8  is  &  .parabola;  no.   9  is  a 
logarithmic  curve;  no.  10  is  a  sinusoid.     Extend  the  curve 
of  no.  10  in  both  directions  as  far  as  the  limits  of  the 


VIII 


GRAPHIC  REPRESENTATION 


113 


paper  will  allow,  using  the  dimensions  of  each  small  ruled 
square  to  represent  a  distance  of  ir/Q  along  each  axis. 


X 

y 

0.0 

.00 

0.2 

.20 

0.4 

.39 

0.6 

.56 

0.8 

.72 

.0 

.84 

.2 

.93 

.4 

.99 

.6 

1.00 

.8 

.97 

2.0 

.91 

2.2 

.81 

2.4 

.67 

2.6 

.62 

2.8 

.34 

3.0 

.14 

3.2 

-.06 

3.4 

-.26 

3.6 

-.44 

3.8 

-.61 

4.0 

-.76 

4.2 

-.87 

4.4 

-.95 

4.6 

-.99 

4.8 

-1.00 

5.0 

-.96 

5.2 

-.88 

5.4 

-.77 

5.6 

-.63 

5.8 

-.46 

6.0 

-.28 

6.2 

-.09 

6.4 

+  .12 

6.6 

.31 

X 

y 

0.0 

.00 

0.2 

.20 

0.4 

.42 

0.6 

.68 

0.8 

1.03 

1.0 

1.56 

1.2 

(2.6) 

1.4 

(5.8) 

1.6 

(-34.) 

1.8 

(-4.) 

2.0 

-2.19 

2.2 

-1.37 

2.4 

-   .92 

2.6 

-   .60 

2.8 

-   .36 

3.0 

-   .14 

3.2 

+  .06 

3.4 

.26 

3.6 

.49 

3.8 

.77 

4.0 

1.16 

4.2 

1.77 

4.4 

(3.1) 

4.6 

(8.9) 

4.8 

(11.) 

5.6 

(3.4) 

5.2 

-1.89 

5.4 

-1.22 

5.6 

-   .81 

5.8 

-  .52 

6.0 

-   .29 

6.2 

-   .09 

6.4 

+   .12 

6.6 

.33 

X 

y 

0.0 

4.00 

0.4 

1.87 

0.8 

1.22 

1.2 

0.82 

1.6 

0.54 

2.0 

0.34 

2.4 

0.20 

2.8 

0.08 

3.2 

0.02 

3.6 

0.01 

4.0 

0.00 

4.4 

0.01 

4.8 

0.04 

5.2 

0.08 

5.6 

0.13 

6.0 

0.20 

6.4 

0.28 

6.8 

0.37 

7.2 

0.47 

7.6 

0.57 

8.0 

0.69 

1.87 

0.4 

1.22 

0.8 

0.82 

1.2 

0.54 

1.6 

0.34 

2.0 

0.20 

2.4 

0.08 

2.8 

0.02 

3.2 

0.01 

3.6 

0.00 

4.0 

0.01 

4.4 

etc. 

etc. 

TABLE  6  TABLE  7  TABLE  8 

The  curves  of  Tables  6  and  7  should  both  be  drawn  on 
a  single  diagram,  and  the  points  Tr/2;  TT,  3ir/2,  and  2ir 
should  be  marked  on  the  #-axis  in  addition  to  the  usual 
scale.  The  curve  for  Table  9  also  may  be  drawn  on  the 
same  diagram;  use  the  ^/-values  as  they  stand. 
9 


114 


THEORY  OF  MEASUREMENTS 


§102 


X 

y 

0.1 

-3.  +  .70 

0.2 

-2.  +  .3Q 

0.3 

-2.  +  .80 

0.4 

-1.  +  .08 

0.5 

-1.  +  .31 

0.6 

-1.  +  .49 

0.7 

-1.  +  .64 

0.8 

-1.  +  .78 

0.9 

-1.  +  .89 

1.0 

O.  +  .OO 

1.2 

0.  +  .18 

1.4 

.34 

1.6 

.47 

1.8 

.59 

2.0 

.69 

2.2 

.79 

2.4 

.88 

2.6 

.96 

2.8 

1.03 

3.0 

1.10 

3.2 

1.16 

3.4 

1.22 

3.6 

1.28 

3.8 

1.34 

4.0 

1.39 

X 

y 

0 

7T/6 

7T/3 
7T/2 
27T/3 
57T/6 
7T 

0 
.95X/6 
1.657T/6 

1.917T/6 

1.657T/6 
.95W6 
0 

7lT/6 
87T/6 
97T/6 

10ir/6 

llTT/6 

2;r 

-  .957T/6 
-1.657T/6 

-1.917T/6 

-1.657T/6 
-  .95W6 
0 

13W6 

14ir/6 

157T/6 
167T/6 
177T/6 

.3x 

+  .957T/6 
1.65W6 

1.917T/6 

1.657T/6 
.957T/6 
0 

197T/6 

etc. 

-  .95W6 
etc. 

-7T/6 

-7T/3 
-7T/2 

etc. 

-   .95W6 
-1.657T/6 

-1.917T/6 

etc. 

TABLE  9 


TABLE  10 


IX.     CURVES  AND   EQUATIONS 

Apparatus. — A    pencil    with    a    sharp    point;    pencil 
compass. 

102.  Graphic   Representation   of   a    Natural  Law. — 

The  graphic  diagram  which  is  obtained  from  a  table  of 
measurements  or  experimental  data  usually  shows  ir- 
regularities, which  are  sometimes  retained  by  the  use  of 
a  broken  line  and  are  sometimes  eliminated  by  the  process 
known  as  " smoothing"  (§  100).  Neither  of  these  pro- 


FIG.  24.     PARABOLIC  PATH  OF  A  FALLING  BODY. — The  total  vertical 
movement  is  proportional  to  the  square  of  the  horizontal  movement. 

cedures  is  necessary  if  the  variables  follow  some  definite 
natural  law  in  regard  to  their  changes  or  if  their  relation- 
ship can  be  expressed  by  an  equation.  In  such  cases  the 
plotted  points  in  general  will  lie  precisely  along  a  smooth 
curve  without  showing  any  irregularities  whatever.  For 
example,  a  ball  that  is  thrown  horizontally  will  travel, 

115 


116  THEORY  OF  MEASUREMENTS  ,      §103 

under  the  influence  of  gravity,  along  a  curved  path  in 
such  a  way  that  its  progress  in  a  horizontal  direction  during 
1,  2,  3,  4,  '  • «  n  seconds  will  be  proportional  to  the  num- 
bers 1,  2,  3,  4,  •  •  •  n,  while  its  downward  movement  will 
be  proportional  to  the  numbers  1,  4,  9,  16,  •  •  •  n2.  If 
points  are  so  located  on  a  graphic  diagram  that  the  ver- 
tical distance  of  each  below  the  z-axis  is  proportional  to 
the  square  of  its  distance  to  the  right  of  the  ?/-axis 
(Fig.  24)  it  will  be  found  that  a  smooth  curve  can  be 
drawn  so  as  to  pass  exactly  through  all  of  them,  and  the 
relationship  between  the  z-value  and  the  ?/-value  of  each 
and  every  point  will  of  course  be  given  by  the  equation 
y  =  —  kx2  (§  10).  Such  a  curve  is  shown  more  or  less 
steadily  by  the  surface  of  water  that  forms  a  waterfall 
or  by  a  jet  of  water  that  issues  from  a  hose  pipe,  and  is 
known  as  a  parabola. 

It  will  be  seen  that  the  scales  have  been  so  chosen  in 
Fig.  24  that  k  =  1/5,  in  other  words  the  equation 
y  =  —  kx2  has  become  y  =  —  \x2.  Test  the  diagram 
by  assuming  that  x  has  the  value  of  3/2,  finding  what  the 
corresponding  value  of  y  must  be  by  solving  the  equation, 
and  then  locating  the  point  (3/2,  •-  9/20)  which  has 
these  two  numbers  for  its  z-value  and  ?/-value.  Does 
this  point  lie  on  the  same  smooth  curve  as  the  points 
that  are  shown?  Do  'the  same  way  with  each  of  the 
z-values  1/2,  5/2,  and  7/2. 

103.  Graph  of  an  Equation. — A  single  equation  that 
contains  two  unknown  quantities  has  an  infinite  number 
of  solutions,  for  any  value  whatever  may  be  assigned  to 
one  unknown  quantity  and  the  corresponding  value  can 
always  be  calculated  for  the  other.  Any  such  solution 
of  an  equation  that  involves  x  and  y  will  consist  of  an 
z-value  and  a  ?/-value,  and  so  can  be  represented  by  the 
position  of  a  point.  All  of  the  infinite  number  of  solii- 


IX 


CURVES  AND  EQUATIONS 


117 


tions  of  a  given  equation  of  this  sort  can  theoretically  be 
represented  by  an  infinite  number  of  points  on  a  graphic 
diagram;  thus  every  point  that  lies  on  the  curve  of  Fig.  24 
corresponds  to  two  numbers,  z-^alue  and  ?/-value,  which 
are  a  solution  of  the  equation  y  =  —  ^x2.  In  general,  all 
of  the  points  that  represent  the  solutions  of  a  given  equa- 
tion will  be  found  to  lie  along  a  smooth  curved  or  straight 
line,  which  is  accordingly  called  the  locus  or  "curve  "  of  the 
equation.  If  the  locus  of  an  equation  extends  to  an 
infinite  distance,  as  is  the  case  with  the  curve  of 
y  =  —  ^x2,  it?  distant  portions  are  usually  more  or  less  flat 
or  straight  and  uninteresting,  so  that  there  is  no  objection 
to  omitting  them  from  the  graphic  diagram. 


X 

y 

0 

0 

1 

0 

2 

6 

3 

24 

4 

60 

5 

120 

-1 

0 

-2 

-  6 

-3 

-24 

TABLE  OF  SOLU- 
TIONS OF  THE  EQUA- 
TION y  =  x3  —  x. 
The  ^/-values  in- 
crease so  rapidly 
beyond  x  =  5  that 
the  curve  must  be 
nearly  straight. 


FIG.  25.  GRAPH  OF  THE  EQUATION  y  =  x*  —  x.  Notice  that 
the  points  (—  1,0),  (0,  0),  and  (+  1,  0)  are  not  close  enough  together 
to  determine  the  shape  of  the  curve  without  plotting  additional 
points. 

It  should  be  carefully  kept  in  mind  that  the  relation- 
ship between  an  equation  and  its  " curve"  or  locus  means 


118  THEORY  OF  MEASUREMENTS  §104 

that  every  solution  of  the  equation  gives  a  point  which  is 
located  on  the  curve,  and  that  every  point  on  the  curve 
furnishes  a  solution  of  the  equation.  From  these  two 
statements  it  is  evident  that  the  coordinates  of  a  point 
which  is  not  on  the  curve  will  not  satisfy  the  equation 
(try  the  point  (4,  —  9/5)  shown  by  the  letter  P  in  Fig. 
24),  and  that  an  incorrect  solution  (try  x  =  2.  y  =  +  4/5) 
will  give  a  point  which  does  not  lie  on  the  curve. 

It  is  obviously  impossible  to  calculate  the  position  of 
an  infinite  number  of  points,  but  since  an  equation  is 
known  to  give  a  smooth  curve  it  is  only  necessary  to  plot 
enough  points  to  prevent  uncertainty  as  to  the  shape  and 
position  of  any  part  of  it. 

Make  a  graphic  diagram  for  the  equation  y  =  x3  —  x 
by  the  following  process :  Assume  that  x  has  various  small 
integral  values,  positive  and  negative,  and  calculate 
the  corresponding  values  of  y,  arranging  them  in  tabular 
form,  as  shown  on  page  117.  The  z- values  have  not 
been  carried  beyond  +  5  on  account  of  the  ^/-values 
increasing  so  rapidly.  Negative  ^-values  are  seen  to 
have  the  same  numerical  ^-values  as  positive  ones  but 
with  the  sign  changed.  The  next  step  is  to  choose 
suitable  axes  and  scales  (§  98).  Plot  the  points  given 
by  the  table  before  proceeding  further.  It  is  not  prob- 
able that  the  line  has  a  straight  portion  between  (1,  0) 
and  '(—  1,  0)  while  the  rest  of  it  is  curved,  so  it  is  neces- 
sary to  calculate  at  least  two  more  points  on  the 
curve,  e.  g.,  x  =  1/2  and  x  =  —  1/2;  and  these  will  be 
found  sufficient  to  determine  a  smoothly  flowing  curved 
outline. 

104.  General  Procedure. — In  plotting  the  curve  of  an 
equation  the  general  procedure  is  to  calculate  the  table 
of  values,  substituting  successive  positive  and  negative 
integral  values  of  x  (and  fractional  values  if  necessary) 


IX  CURVES  AND  EQUATIONS  119 

and  solving  for  y,  then  consider  the  available  space  on 
the  paper  and  draw  the  axes  in  a  suitable  location. 
Before  locating  the  points  that  correspond  to  the  tabular 
values  a  scale  of  numbers  should  always  be  marked  along 
the  a>axis  and  another  along  the  i/-axis.  The  two  scales 
need  not  be  the  same  (§  97),  but  along  each  axis,  con- 
sidered by  itself,  equal  distances  must  always  correspond 
to  equal  numerical  differences,  and  z-values  must  always 
increase  to  the  right  and  ?/-values  increase  upward.  If  a 
condensed  scale  is  used  it  is  always  advisable  to  let  the 
dimensions  of  a  small  square  of  the  ruled  paper  represent 
a  simple  round  number;  if  the  first  tabular  value  is 
9200  do  not  number  the  successive  squares  9200,  18400, 
27600,  etc.,  but  use  the  simpler  round  numbers  10000, 
20000,  30000.  Where  the  relative  value  of  the  z-unit 
as  compared  with  the  y-umt  is  unimportant  the  scales 
can  often  be  advantageously  chosen  so  as  to  give  the 
chief  part  of  the  curve  a  slope  of  about  45°.  In  the  rest 
of  the  exercises  of  this  chapter,  however,  a  single  square 
of  the  paper  is  to  represent  a  single  unit,  in  each  direc- 
tion, except  where  otherwise  specified. 

105.  The  Straight  Line. — The  '  equation  y  =  2x  -f  3 
represents  a  sloping  straight  line.  For  such  a  simple 
equation  it  is  hardly  necessary  to  construct  a  table  of 
values.  Locate  the  x-axis  and  ?/-axis  in  your  notebook 
where  there  is  room  for  the  y-values  to  extend  approxi- 
mately from  +  10  to  —  10  and  for  the  x- values  to 
extend  at  least  to  ±5..  Plot  four  or  five  points  from 
solutions  of  y  =  2x  +  3  obtained  mentally,  and  use  a 
ruler  to  draw  a  straight  line  passing  through  them  and 
extending  a  little  distance  beyond  them  in  each  direc- 
tion. Label  this  line  by  writing  the  equation  y  =  2x  +  3 
alongside  it,  close  to  one  end.  Without  making  a  new 
diagram  use  the  same  axes  and  scales  for  plotting  the 


120  THEORY  OF  MEASUREMENTS  §105 

following  straight  lines:  y  =  3  —  2x;  y  =  3  -f-  Jz;* 
y  =  3  +  Oz;  y  =  2z;  y  =  -  1  +  x;  y  =  -  1  +  Jz; 
2/  =  —  1  —  2#;  ?/  =  —  1  —  x.  Label  each  of  these  in 
the  same  way.  Examine  the  lines  whose  equations  have 
3  for  the  numerical  term.  Do  they  all  pass  through  the 
point  (0,  3)?  Do  the  lines  of  the  equations  that  have 
—  1  for  a  numerical  term  all  pass  through  the  point 
(0,  —  1)?  Which  line  passes  through  the  origin  (0,  0)? 
Does  it  seem  probable  that  the  equation  y  =  a  +  bx 
gives  a  line  that  passes  through  the  point  (0,  a)?  Test 
the  point  (0,  a)  by  ordinary  algebra  in  order  to  deter- 
mine whether  it  lies  on  the  line  y  =  a  +  bx. 

Determine  the  slope  (§  38  or  17)  of  the  line  which 
you  have  drawn  to  represent  the  equation  y  =  2#  -f  3, 
and  notice  that  if  the  z-value  of  one  point  on  the  line  is 
one  unit  greater  than  that  of  another  point  on  the  line 
then  the  ^/-value  is  always  two  units  greater;  furthermore, 
for  any  two  points  on  the  line  the  difference  in  ^/-values 
is  twice  as  great  as  the  difference  in  ^-values.  If  the 
difference  in  z-values  is  denoted  by  Ax  and  the  corre- 
sponding difference  in  ^/-values  by  Ay  the  last  statement 
can  be  written  in  the  form  of  an  equation  Ay  =  2Ax  or 
Ay/Ax  =  2.f 

In  the  line  representing  the  equation  y  =  3  +  %x 
does  a  unit  change  in  x  correspond  to  a  change  of  J  in  y? 
In  y  =  3  +  Ox  does  a  unit  change  in  x  cause  no  change 
in  y!  In  y  =  3  —  2x  does  a  unit  change  in  x  cause  an 
increase  of  —  2  units  in  y?  Can  the  slope  of  this  last 
line  be  considered  as  equal  to  —  2?  Prove  algebra- 

*  Notice  that  this  equation,  as  it  stands,  is  an  explicit  statement 
of  the  value  of  y,  "  clearing  of  fractions  "  before  starting  to  substitute 
would  only  complicate  the  work. 

t  In  this  notation  the  symbol  A  is  not  used  to  denote  an  individual 
quantity  or  factor  but  is  a  "symbol  of  functionality"  like  the  "log" 
or  "cos"  of  the  expressions  log  x  and  cos  x. 


IX 


CURVES  AND  EQUATIONS 


121 


A 


ically  that  when  x  increases  by  one  unit    (e.   g.,  when 

it  changes  from  m  to  m  +  1)  the  value  of  y  iii  the 

equation  y  =  a  +  bx  will  increase 

by  b  units;  thus  showing  that  the 

equation  represents   a  line  whose 

slope   is   numerically   equal    to    b. 

From  this  it  is  evident  that  Ay/ Ax 

=  b  for  the  line  y  =  a  +  bx,  and 

that  the  coefficient  of  x  gives  the  slope 

of  a  line  if  its  equation  is  arranged  in 

the    form    y  =  a  +  bx.      Turn    to 

your  diagram  and  see  whether  the 

lines  y  =  3  —  2x  and  y  =  —  1  —  2x 

are  parallel    (i.  e.,  have   the  same 

slope).     The  equation  Ay  /Ax  =  —  2 

is  true  of  both  of  them  and  of  any 

other  line   that  runs   in  the  same 

direction.     Just   as  the   line  y  =  3 

—  2x  represents  all  of  the  infinitely 

numerous  points  that  lie  along  its 

locus   (§  103)   and  has   an  infinite 

number  of  solutions,  so  the  still  more 

general   relationship    Ay/Ax  =  —  2 

represents    an   infinite   number   of 

parallel  lines,    any    one   of  which 

may  be  considered  as  a  solution  of 

it. 

The  equation  y  =  a  +  bx  is  the  general  equation  for 
all  straight  lines  except  those  that  run  parallel  to  the 
i/-axis.  The  latter  are  given  by  the  equation  x  =  k. 
(Plot  the  equation  x  =  2  by  substituting  integral  values 
of  y  instead  of  x,  and  solving  for  x  instead  of  y.) 
The  most  general  equation  of  the  straight  line  is 
Ax  +  By  +  C  =  0.  This  can  be  reduced  to  the  form 


A'x 

FIG.  26.  THE 
STRAIGHT  LINE. —  If 
Pi,  PZ,  and  PS  are  any 
points  on  the  (curved 
or  straight)  line  whose 
equation  is  y  =  a  +  bx 
it  is  easy  to  show  alge- 
braically that  Ay/Ax 
=  A'y/A'z,  and  thence 
geometrically  (by  simi- 
lar triangles)  to  prove 
that  the  "curve"  is  a 
straight  line. 


122  THEORY  OF  MEASUREMENTS  §106 

y  =  a-{-bx  if  B  is  different  from  zero,  and  to  the  form 
x  =  k  if  B  is  zero. 

106.  The  Parabola. — On  a  single  graphic  diagram 
draw  and  label  the  curves  y  =  x2,  y  =  x2  +  3,  y  =  —  x2, 
and  y  =  x2/lO.  The  first  three  of  these  need  not  extend 
beyond  x  =  =t  3,  but  the  last  one  should  be  drawn  from 
x  =  —  10  to  x  =  +10.  They  are  all  similar  figures  and 
are  called  parabolas;  the  curves  y  =  x2  and  y  =  x2/lO 
differ  in  size  but  the  portion  of  the  latter  that  extends 
from  —  10  to  +  10  is  of  exactly  the  same  shape  as  the 
part  of  the  former  that  is  included  between  x  =  —  1  and 
x  =  +  1.  Notice  that  the  curve  y  =  x2  +  3  differs 
only  in  position  from  y  =  x2;  it  appears  to  be  the  same 
curve  moved  three  units  higher  up,  and  the  equations 
make  it  clear  that  for  each  z-value  the  ^/-value  of  the 
first  curve  is  greater  by  three  than  that  of  the  second. 

On  another  graphic  diagram  draw  and  label  the  three 
curves  y  =  x2  —  3x,  y  =  x2  —  6x,  and  y  =  x2  —  7x. 
Notice  that  they  are  not  only  of  the  same  shape  but  of 
the  same  size  as  well. 

The  general  equation  of  all  parabolas  whose  axis  of 
symmetry  is  vertical  is  y  =  a  +  bx  +  ex2.  The  size  of 
the  curve  depends  only  upon  the  value  of  c  and  it  is 
convex  downward  (" festoon-shaped")  if  c  is  positive, 
convex  upward  (" arch-shaped")  if  c*  is  negative.  A 
change  in  the  value  of  a  has  been  shown  to  raise  (or 
lower)  the  curve  through  a  corresponding  distance,  and 
obviously  it  always  cuts  the  ?/-axis  at  a  height  of  a. 
Since  any  curve  cuts  the  z-axis  at  the  points  where 
y  =  0  .it  follows  that  the  parabola  y  =  a  +  bx  +  ex2 
must  cross  the  z-axis  at  points  whose  abscissae  are  the 
two  roots  of  the  quadratic  equation  0  =  a  +  bx  +  ex2; 
these  roots  of  course  depend  upon  the  values  of  all  three 
of  the  parameters  a,  6,  and  c. 


IX  CURVES  AND  EQUATIONS  123 

107.  The  Probability  Curve. — In  the  chapter  on  loga- 
rithms directions  were  given  for  calculating  the  value  of 
y  =  e~x2    when    x    assumes    various    positive    values.* 
Turn  to  the  table  in  your  notebook  that  gives  the  cor- 
responding values  of  x  and  y  for  this  equation  and  plot 
its  graphic  diagram  on  the  vacant  page  next  to  it.     Turn 
the  notebook  so  that  the  x-axis  will  lie  lengthwise  of  the 
page  and  use  as  large  a  scale  as  will  conveniently  permit 
the  x-values  to  extend  from  —  0.5  to  +  2.5  or  3  (say  20 
squares  =  1    unit,    both    horizontally    and    vertically). 
For  negative  values  of  x  notice  that  the  curve  is  symmet- 
rical with  respect  to  the  x-axis;  that  is,  if  one  point  on 
the  curve  is  (+  0.5,  +  0.78)  or  (a,  b)  then  another  one 
will  be    (-0.5,    +0.78)    or   (-a,    +6).     This   could 
have  been  inferred  from  the  equation  y  =  e~^,  for  x 
occurs  in  it  only  as  an  even  power  and  any  function  of 
(—  a)2  must  have  the  same  value  as  the  same  function  of 

(+  a)2- 

108.  Equation  of  a  Graph. — The  process  of  finding  the 
curve  that  corresponds  to  a  given  equation  is  not  usually 
difficult,  but  the  reverse  operation  may  be  much  harder. 
If  the  given  " curve"  is  a  straight  line  that  cuts  the  7/-axis 
its  equation  will  be  of  the  form  y  =  a  +  bx,  and  the 
values  of  a  and  b  can  be  determined  according  to  the 
two  italicized  propositions  in  §  105. 

Draw  a  line  through  the  two  points  (1,  3)  and  (2,  1), 
and  find  its  equation.  Ans.:  y  =  5  —  2x.  Draw  a  line 
through  (0,  2)  and  (3,  0),  and  find  its  equation.  Find 
the  equation  of  a  line  drawn  through  (0,  —  3)  and 

*  Although  an  equation  is  properly  a  sentence  the  student  will 
sometimes  find  it  apparently  used  as  a  noun.  In  such  cases  one 
side  of  the  equation  is  to  be  considered  as  the  substantive  and  all  the 
rest  of  it  as  a  parenthetical  statement;  thus  the  expression  in  the 
text  means,  "for  calculating  the  value  of  y  (which  is  the  same  thing 
as  e~*2)  when  x  assumes.  ..." 


124  THEORY  OF  MEASUREMENTS  §109 

(5,  0).  See  if  your  last  two  equations  can  be  transformed 
into  x/3  +  y/2  =  1  and  x/5  +y/  (-  3)  =  1,  respectively. 
What  is  the  apparent  significance  of  the  denominators  m 
and  n  in  the  equation  x/m  +  y/n  =1? 

If  the  given  curve  is  a  parabola  it  has  been  seen  that 
its  equation  will  be  y  =  mx2  if  the  origin  is  at  the  vertex 
of  the  curve.  Turn  to  your  plot  of  y  =  x2  —  Qx  (§  106) 
and  draw  a  new  pair  of  axes  through  the  point  (3,  —  9), 
which  is  the  vertex  of  the  curve.  If  the  coordinates  of 
points  on  the  curve,  referred  to  these  new  axes,  are 
denoted  by  the  capital  letters  X  and  F,  notice  that  the 
same  curve  corresponds  exactly  to  the  relationship 
F  =  IX2.  From  this  equation  it  is  possible  to  deduce 
the  original  equation  as  follows:  Let  P  be  any  point  on 
the  curve;  notice  that  its  X-value  is  3  less  than  its  z-value, 
also  that  its  F-value  is  9  more  than  its  y-value.  That 
is,  for  every  point  on  the  curve  the  equations  X  =  x  —  3 
and  F  =  y  +  9  hold  good,  as  well  as  F  =  IX2.  Sub- 
stituting in  the  last  equation  the  values  of  X  and  F  given 
by  the  other  two,  y  +  9  =  (x  —  3)2  or  y  =  x2  —  Qx. 

Find  the  equation  of  the  parabola  that  was  drawn 
for  Table  A,  §  96. 

With  a  pencil  compass  draw  carefully  a  circle  whose 
radius  is  5  and  whose  centre  is  the  point  (4,  3)  on  a  graphic 
diagram.  The  curve  will  be  seen  to  pass  exactly  through 
the  points  (0,  0),  (1,  —  1),  (4,  —  2),  and  three  corre- 
sponding integral  points  in  each  quadrant.  Mark  any 
point  P  on  the  circumference,  and  considering  the  centre 
of  the  circle  as  a  new  origin  draw  the  ordinate  F  of  the 
point  P  and  from  its  base  draw  the  abscissa  X  along  the 
new  X-axis.  From  the  theorem  of  Pythagoras  and  the 
fact  that  all  radii  are  equal  the  coordinates  of  the  point 
P  must  satisfy  the  relationship  X2  -f-  F2  =  52;  and 
since  X  =  x  —  4  and  F  =  y  —  3  this  relationship  be- 


IX  CURVES  AND  EQUATIONS  125 


comes  (x  -  4)2  +  (y  -  3)2  =  52,ory  =  3  +1/25 -(x- 4)2, 
which  is  the  equation  of  the  given  circle. 

109.  Change  of  Scales. — Lay  off  an  z-scale  in  which 
three  squares  of  the  ruled  paper  correspond  to  each  unit 
and  a  ?/-scale  of  two  squares  per  unit,  and  plot  the  circle 
x2  +  y2  =  25  from  the  following  twelve  points  (0,  =b  5), 
(=b  3,  ±  4),  (±  4,  ±  3),  (zb  5,  0),  drawing  as  smooth  a 
curve  as  possible.     The  result  is  an  ellipse,  or  a  "  strained" 
(i.  e.t  distorted)  circle. 

To  find  the  equation  of  such  an  ellipse,  referred  to 
uniform  scales,  let  each  square  of  the  ruled  paper  be 
considered  as  equal  to  unity.  Then  each  point  on  the 
ellipse  will  be  twice  as  high  or  low  and  three  times  as 
far  to  the  right  or  left  as  a  corresponding,,  point  on  the 
true  circle  x2  -f  y'2  =  25.  That  is,  if  the  coordinates  of  a 
point  on  the  ellipse  are  called  X  and  F,  then  X  =  3x 
and  Y  =  2y,  in  other  words,  J-  of  the  X-value  and  J  of 
the  F-value  will  satisfy  the  relationship  for  the  circle,  so 
that  (X/3)2  +  (F/2)2  =  25.  This  is  an  illustration  of 
the  general  fact  that  re- writing  an  equation  with  xfa  and 
y/b  in  place  of  the  original  x  and  y  will  stretch  out  the 
length  and  height  of  the  diagram  to  a  and  6  times  the 
original  dimensions,  respectively. 

On  one  graphic  diagram  plot  the  loci  of  x  +  y  =  1  and 
Z/3  +  1//5  =  I.' 

110.  Definitions  of  Circular  Functions. — The  use  of  a 
graphic  diagram  makes  it  possible  to  give  very  concise 
definitions  of  the  circular  functions  of  an  angle  of  any 
size:  Let  a  line  that  coincides  with  OX  (Fig.  27)  rotate 
counterclockwise  through  an  angle  a  to  a  new  position 
OP.     If  the  coordinates  of  the  point  P  are  denoted  'by 
x  and  y,  and  its  distance  from  the  origin  by  +  r,  then 
sin   a  =  y/r,    cos    a  =  x/r,    tan   a  =  y/x,    cot   a  =  x/y, 
sec  a  =  r/x,  and  esc  a  =  r/y.     These  relationships  hold 


126 


THEORY  OF  MEASUREMENTS 


§111 


true  whether  the  angle  is  larger  or  smaller  than  360°. 
and  for  negative  angles  when  described  by  a  clockwise 
rotation.  They  furnish  the  easiest  method  of  determin- 
ing the  sign  of  a  circular  function;  in  Fig.  28  (<  103°)  the 


270 


FIG.  27.  FUNCTIONS  OF  AN  ANGLE. — For  an  angle  of  any  magni- 
tude, described  counterclockwise  from  OX,  the  sine,  cosine,  tangent, 
cotangent,  secant,  and  cosecant  are  respectively  y/r,  x/r,  y/x,  x/y, 
rfx,  and  r/y,  where  r  is  the  distance  from  the  point  (x,  y)  on  the 
rotating  line  to  the  origin. 

value  of  y  is  positive  but  x  is  negative;  r  is  always  con- 
sidered positive,  so  it  is  immediately  obvious  that  the 
sine  of  an  angle  "in  the  second  quadrant"  (between  90° 
and  180°)  is  positive  while  the  cosine  is  negative. 

111.  Questions  and  Exercises. — 1.  If  one  point  on  the 
curve  of  y  =  x3  —  x  (Fig.  25)  is  (a,  6)  prove  that  another 
one  is  (—  a,  —  b). 

2.  Knowing  the  shape  of  the  graph  of  y  =  a-\-bx+cx2, 
what  facts  can  you  deduce  in  respect  to  the  graph  of 
x  =  a  +  by  +  q/2? 

3.  What  effect  will  be  produced  if  the  equation  of  a 
curve  is  re-written  with  x  +  p  and  y  -\-  q  substituted 
everywhere  for  the  original  x  and  yl     (Try  the  parabola, 


IX  CURVES  AND  EQUATIONS  127 

y  =  x2,  and  p  =  0,  q  =  3,  if  there  is  any  uncertainty.) 

4.  What  effect  will  be  produced  if  the  equation  of  a 
curve  is  re-written  with  mx  substituted  everywhere  for 
the  original  #?     If  my  for  the  original  i/? 

5.  Make  a  graphic  diagram  of  the  equation  y  =  30/x. 
Plot  enough  points  to  show  clearly  the  form  of  both  parts 
of   the   cur  ye.     On   the   same    diagram   plot   y  =  l/x. 
Each  of  these  curves  is  a  (rectangular)  hyperbola. 

6.  Plot  the  curve  of  y  =  l/x2  carefully.     How  would 
the  appearance  of  the  curve  y  =  30/x2  differ? 

7.  Solve  graphically  the  two  simultaneous   equations 
x2  -\-  y2  =  25  and  y  =  x2  —  3x  by  drawing  both  curves  on 
one  diagram  and  locating  their  points  of  intersection. 

8.  Plot  the  curves  of  one  or  more  of  the  following 
equations:    (a)    the    "semi-cubical"    parabola    y2  =  xs; 
(b)  the  finite  curve  y2  =  x(W  —  x)3,  using  a  condensed 
scale  along  the  ?/-axis;  (c)  the  curve  y  =  x  log  x]  (d) 

X2    _   y2    =    Q.    (e)    X2    +   02  ,=    0. 

9.  Instead  of  being  located  by  latitude  and  longitude 
(rectilinear  coordinates,  referred  to  an  z-axis  and  a  ?/-axis), 
the  position  of  a  point  may  be  located  by  its  distance  and 
direction,  i.  e.,  the  length  r  and  the  angle  a  of  §  110  (its 
polar  coordinates,  referred  to  a  pole  and  an  initial  line). 
Obviously,  tan  a  =  y/x  and  r2  =  x2  +  y2}  or  x  =  r  cos  a 
and  y  =  r  sin  a.     The  distance  and  angle  are  called  polar 
coordinates  and  are  usually  indicated  by  the  letters  r 
(radius)  and  6  (angle),  or  by  p  and  0. 

Make  a  rough  graphic  diagram  of  the  curve  of  the 
equation  x2  +  (y  —  h)2  =  h2  (a  circle  that  passes  through 
the  points  (0,  0),  (0,  2h),  and  (±  h,  +  h);  compare  §  108); 
then  substitute  x  =  p  cos  B,  y  =  p  sin  6  and  show  that  the 
polar  equation  of  this  circle  is  p  =  2h  sin  6. 

What  must  be  the  general  shape  of  the  curve  p  =  20? 

Of  (a)  p  =  2;  (b)  p  =  I/cos  e  or  p  =  sec  6',  (c)  6  =  1. 


X.     GRAPHIC   ANALYSIS 


47 


9 


Apparatus.  —  Fine  black  silk  thread;  slide  rule;  gradu- 
ated ruler;  pencil  with  a  sharp  point. 

112.  Interpretation  of  Equations.  —  Turn  back  to  the 
diagrams  that  were  drawn  to  represent  the  equations 
y  =  2x  and  y  =  2x  +  3  (§  105)  and  notice  that  for  every 
point  of  the  first  there  is  another  point  just  three  units 
higher  on  the  second  of  the  two  straight  lines.  The 

equation  y  =  2x  +  3  makes 
the  statement  that  the  value 
of  y  Is  as  much  as  the  2x  of 
the  first  equation  with  three 
more  added.  Another  way  of 
looking  at  the  same  equation 
can  be  shown  by  writing  the 
terms  in  reversed  order,  y  =  3 
-f-  2x.  This  can  be  considered 
as  making  the  statement  that, 
for  every  value  of  x,  y  is  equal 
to  the  amount  3,  to  which  there 
is  further  added  the  amount 
2x  (Fig.  28).  It  is  worth 
while  to  form  the  habit  of 
always  investigating  the 
meaning  of  an  equation  as 
far  as  possible,  especially  for 
the  student  who  intends  to 
proceed  further  with  the 
study  of  any  of  the  practical 
applications  of  mathematics.  For  example  the  equa- 
tion, s  =  vot  +  %at2,  of  uniformly  accelerated  motion 

128 


FIG.  28.  GRAPH  OF  THE 
EQUATION  y  =  3  +  2x. — The 
ordinate  of  any  point  (D)  on 
the  line  may  be  considered  as 
being  either  3  more  than  2x 
(CD  added  on  to  AC}  or  2x 
more  than  3  (BD  added  on 
toAB). 


X  GRAPHIC  ANALYSIS  129 

means  that  the  space  traversed  is  equal  to  the  space, 
vrf,  that  would  have  been  traversed  by  an  object  moving 
at  an  initial  velocity  of  VQ  without  acceleration,  plus  the 
\a&  of  distance  that  a  stationary  body  would  have  been 
made  to  describe  if  acted  upon  by  the  accelerating  force 
alone — an  illustration  of  the  addition  of  vectors,  which 
in  this  case  are  directed  along  the  same  straight  line. 
The  student  of  physics  will  find  no  difficulty  in  further 
analyzing  each  of  these  two  separate  terms. 

113.  The  Graph  of  y  =  a  +  bx. — It  has  already  been 
seen  (§  105)  that  the  equation  y  =  a  +  bx  has  a  straight 
line  for  its  "  curve,"  and  that  this  line  cuts  the  ?/-axis 
at  a  height  of  a  and  has  a  slope  that  is  numerically  equal 
to  b.  For  this  reason  it  is  often  spoken  of  as  a  linear 
equation,  and  the  law  of  variation  which  it  illustrates  is 
known  as  the  "straight  line"  law.  It  should  be  noticed 
that  the  values  of  x  and  y  are  not  proportional  except 
for  those  cases  in  which  the  straight  line  passes  through 
the  origin. 

Draw  the  loci  of  the  following  equations  without 
calculating  the  values  of  x  and  y  for  any  point.  For  each 
one  rule  a  straight  line  in  such  a  position  that  it  inter- 
sects the  y-axis  at  the  required  point  and  has  the  re- 
quired gradient.  If  there  is  any  doubt  as  to  the  meaning 
of  a  negative  value  for  the  b  of  the  equation  y  =  a  +  bx 
a  few  points  may  be  calculated  from  the  equation  (5) 
in  the  usual  manner.  (1)  y  =  2  +  2x.  (2)  y  =  2  +  x. 
(3)  y  =  2  +  Jx.  (4)  y  =  2  +  Qx.  (5)  y  =  2  --  \x. 
(6)  y  =  4  -  2x.  (7)  y  =  4  -  x.  (8)  y  =  0  -  J*. 
(9)  y  =  \x.  (10)  y  =  -  2  -  x.  (11)  y  =  -  2  +  x. 
(12)  y  =  -  2. 

Lay  the  ruler  on  the  squared  paper  at  random  in  any 
position  and  draw  a  straight  line.  Mark  the  axes  along 
any  convenient  ruled  lines  of  the  paper  and  determine  the 
10 


130  THEORY  OF  MEASUREMENTS  §114 

equation  of  the  line  that  was  drawn,  expressing  the  nu- 
merical term  and  the  coefficient  of  x  as  decimal  fractions. 
1 14.  The  Straight  Line  Law. — If  a  homogeneous  metal 
bar  is.  heated  it  may  be  expected  to  expand  in  such  a  way 
that  its  length  is  always  proportional  to  its  thickness,  and 
if  these  two  variables  are  denoted  by  x  and  y  the  relation 
between  them  will  be  given  by  the  equation  y  =  kx,  in 
which  k  has  a  constant  value.  If  we  compare  length  and 
temperature,  however,  there  is  no  proportionality  be- 
tween them.  The  bar  is  not  made  twice  as  long  by 
heating  it  twice  as  hot,  but  it  is  not  difficult  to  show  ex- 
perimentally that  there  is  certain  law  of  relationship, 
namely,  -the  change  in  length  is  proportional  to  the 
change  in  temperature.  If  a  bar  whose  length  is  10  at  a 
temperature  of  15°  C.  is  expanded  to  12  at  20°  C.,  then 
its  length  at  30°  C.  will  be  16.*  In  other  words, 
12-10  :  16-10  ::  20-15  :  30-15.  Draw  a  graphic  dia- 
gram to  illustrate  this  example,  plotting  temperature 
horizontally  (§  98)  and  length  vertically,  and  notice  that 
the  points  (15,  10),  (20,  12),  and  (30,  16)  lie  in  the  same 
straight  line.  The  variables  x  and  y  are  not  proportional ; 
it  is  only  their  respective  changes  or  differences  that 
show  proportionality.  Using  the  notation  of  §  105,  If  2, 
we  can  write  AT/  :  Ax  : :  A'y  :  A'x,  or  AI//AX  =  k  =  2/5 
for  the  graph  (draw  these  A's  on  your  diagram),  and  for 
the  general  physical  law  A(length)/A.(temp.)  =  k.  When 
two  variables  show  proportional  changes  they  are  said 
to  follow  a  linear  law,  or  a  straight  line  law. 

*  These  numbers  are  very  greatly  exaggerated  as  compared  with 
those  for  ordinary  materials.  Metals  expand  only  a  few  hundred- 
thousandths  of  any  dimension  when  heated  a  few  degrees.  Further- 
more the  law  is  only  approximate;  careful  experiments  on  a  given 
bar  will  show  that  its  length  at  t°  C.  will  not  be  exactly  expressible 
in  the  form  I  =  J0(l  +  kt),  but  will  need  a  more  complicated  equa- 
tion, I  =  Jo(l  +  M.+  W)  or  even  I  =  I0(l  +  kit  + 


GRAPHIC  ANALYSIS 


131 


Prove  (a)  that  proportionality  is  always  in  accordance 
with  the  straight  line  law,  but  (6)  that  a  straight  line  law 
does  not  mean  proportionality  between  the  two  variables 
except  when  the  straight  line  passes  through  the  origin. 

115.  The  "Black  Thread"  Method.— If  a  set  of  ex- 
perimental measurements,  such  as  those  of  temperature 
and  length  of  a  metal  bar,  are  found  to  correspond  ap- 
proximately to  a  straight-line  law  they  may  be  plotted 
as  the  x's  and  y's  of  a  graphic  diagram,  and  their  irregu- 
larities may  be  eliminated  (" smoothed")  by  drawing 
the  straight  line  that  appears  to  come  closest  to  all  of 
the  points.  This  is  called  the  black  thread  method 
because  the  position  of  the  line  is  decided  by  making  use 
of  a  stretched  thread  instead  of  a  ruler;  the  thread  and 
the  points  can  all  be  seen  at  the  same  time,  while  a  ruler 
would  hide  half  of  the  points  if  properly  placed. 

Plot  the  values  given  in  the  table  as  accurately  as 
possible,  marking  each  point  by  a  minute  dot  surrounded 
by  a  small  circle,  or  by  a  cross  com- 
posed of  a  short  vertical  mark  to  in- 
dicate the  exact  value  of  the  abscissa 
and  a  short  horizontal  line  at  the 
exact  height  of  the  ordinate. 

Be  sure  that  the  page  of  the  note- 
book rests  in  a  perfectly  flat  position 
and  stretch  a  fine  black  silk  thread  on 
it  in  such  a  position  that  it  follows  the 
general  direction  of  the  points.  Move 
it  a  trifle  toward  the  top  or  bottom  of 
the  page,  also  rotate  it  slightly,  both 
clockwise  and  counter-clockwise.  At- 
tempt to  get  it  into  such  a  position  that 
it  lies  among  the  points  like  a  smoothed 
curve  (§  100),  following  their  general  trend  but  not 


X 

y 

1 

9.8 

2 

8.5 

3 

8.0 

4 

7.2 

5 

6.7 

6 

6.5 

7              6.2 

8              5.5 

9 

5.0 

10 

4.1 

11 

3.9 

12 

3.2 

13 

2.3 

EXPERIMENTAL 
DETERMINATIONS 
OF  LINEAR  VARIA- 
TION. 


132  THEORY  OF  MEASUREMENTS  §116 

necessarily  passing  exactly  through  any  one  of  them. 
See  that  there  are  about  as  many  points  above  the  line 
as  below  it,  but  if  the  high  points  are  more  numerous 
toward  one  end  of  the  thread  and  the  low  ones  toward 
the  other  end  rotate  the  thread  enough  to  remedy  the 
condition.  When  the  thread  is  finally  arranged  in  the 
most  satisfactory  position  do  not  attempt  to  draw  the 
line  but  notice  where  the  thread  cuts  the  z-axis  and  where 
it  cuts  the  y-Sixis.  From  these  two  numbers  calculate 
the  slope  (gradient)  of  the  thread,  noticing  whether  its 
value  is  positive  or  negative.  Write  the  equation  of  the 
line  that  is  indicated  by  the  thread,  making  y  equal  to  a 
numerical  value  plus  a  certain  number  of  times  z;  i.  e., 
write  the  equation  in  the  form  y  =  a  +  bx  (§  105). 

116.  Intercept  Form  of  a  Linear  Equation. — The 
equation  x/m  +  y/n  =  1  must  be  the  equation  of  a 
straight  line,  since  it  is  easily  reducible  to  the  form 
y  =  a  +  bx.  Substitute  zero  for  the  value  of  x  and  notice 
that  the  corresponding  value  of  y  is  n.  Show  likewise 
that  when  y  is  equal  to  zero  x  will  be  equal  to  m.  In 
other  words  the  graph  of  x/m  +  y/n  =  1  passes  through 
the  points  (0,  n)  and  (m,  0)  (compare  §  108,  H  2). 

Draw  the  straight  line  x/(  —  3)  +  y/2  =  1  by  ruling 
a  line  through  (—  3,  0)  and  (0,  2);  then  reduce  the  equa- 
tion to  the  form  y  =  a  +  bx  and  see  whether  the  coef- 
ficients a  and  b  verify  the  ^-intercept*  and  the  gradient 
of  the  ruled  line. 

Since  m  and  n  are  the  x-intercept  and  y-intercept  of  the 
line  x/m  +  y/n  =  1  the  equation  of  a  line  that  cuts 
both  axes  can  be  written  immediately  without  any  cal- 
culation. Write  the  equation  of  your  black  thread  de- 
termination in  this  form. 

*  The  points  at  which  a  locus  cuts  the  z-axis  and  the  ?/-axis  are 
called  its  x-intercept  and  y-intercept,  respectively. 


X  GRAPHIC  ANALYSIS  133 

117.  The   Graph  of  y  =  a  +  bx  +  ex2.— Just  as  the 
curve  y  =  a  +  bx  can  be  considered  as  having  its  or- 
dinate  for  each  value  of  x  built  up  of  the  ordinate  a 
plus  the  ordinate  bx  (§  112,  Fig.  28),  so  the  more  elabo- 
rate equation  y  =  a  +  bx  +  ex2  can   be   considered   as 
representing  a  curve  which  is  made  by  piling  up  the 
parabola  y  =  ex*  upon  the   slanting  line  y  =  a  -\-  bx. 
Curiously   enough   this   also   represents   a   parabola  in 
every  case  in  which  c  is  different  from  zero;  the  slant  of 
the  straight  line  does  not  cause  the  curve  to  be  lop-sided. 

On  a  single  graphic  diagram  plot  both  y  =  —  O.lz2 
and  y  =  3  +  0.5#.  To  the  ordinates  of  the  latter  add 
(or  subtract,  as  the  case  may  require)  the  ordinates  of 
the  former  for  each  integral  value  of  x,  and  draw  as 
smoothly  as  possible  the  resultant  curve  of  the  equation 
2/  =  (3+0.5z)+  (O.I*2). 

If  a  curve  that  is  obtained  from  experimental  measure- 
ments looks  like  a  portion  of  a  parabola  it  is  possible  to 
draw  an  approximate  tangent,  find  its  linear  equation, 
and  then  determine  a  value  of  c  that  will  make  the  equa- 
tion y  =  a  +  bx  -f  ex2  fit  the  given  curve.  An  algebra- 
ical procedure  that  accomplishes  the  same  result  is  to 
measure  the  coordinates  of  some  point  on  the  curve 
(say  (2,  3))  and  substitute  in  the  general  equation  (giving 
3  =  a  +  6X2  +  cX22);  repeating  this  with  two  more 
points  gives  three  equations,  from  which  the  values  of  the 
unknown  a,  6,  and  c  may  be  determined.  For  a  graphical 
procedure  it  is  usually  more  satisfactory  to  complete  a 
free-hand  parabola  as  far  as  its  vertex,  if  that  is  not 
already  present,  and  then  continue  as  in  the  following 
example. 

118.  Law  of  Density- Variation  for  Water. — The  den- 
sity of  water  at  different  temperatures  is  given  in  the 
table.     If  the  density  is  called  y  (§  98)  and  the  temper- 


134 


THEORY  OF  MEASUREMENTS 


§118 


ature  x,  it  is  required  to  find  the  numerical  values  of  the 
coefficients  of  the  equation  y  =  a  +  bx  +  ex2  that  will 
express  the  law  of  variation. 

The  first  step  is  to  make  a  careful  graph.  Use  ex- 
tended scales  for  both  ^-values  and  ^/-values  so  that  the 
diagram  will  cover  practically  a  whole  page  of  your 
notebook,  noticing  that  the  ^/-values  need  not  include 

zero  but  only  extend 
from  .995  near  the  bot- 
tom of  the  page  through 
.996,  .997,  .998,  and  .999 
to  1.000  near  the  top. 
The  curve  will  be  seen 
to  have  the  appearance 
of  an  "  arch-shaped  " 
parabola  (§  106),  so  that 
it  is  evident  that  its 
equation  will  be  approxi- 
mately y  =  —  mx2  if  the 
origin  is  located  at  the 
vertex  of  the  curve, 
namely  at  the  point  (4, 
.99997).  With  the  origin 
in  this  position  the  ordi- 
nate  for  4°  will  obviously 

be  zero,  for  a  temperature  2°  higher  than  this  the  ordinate 
will  be  -.00003  (i.  e.,  .99997  -  .99994),  for  4°  higher  it 
will  be  —  .00012,  etc.  These  ^/-values  and  ^-values  have 
been  given  in  the  second  table,  where  for  simplicity  the 
negative  signs  and  the  decimal  points  have  been  omitted. 
If  the  values  in  the  second  table  correspond  to  an 
equation  of  the  form  y  =  kx2  the  square  root  of  y  must 
be  proportional  to  x  itself.  Using  the  slide  rule,  read 
off  the  square  roots  of  the  numbers  in  column  y  and  enter 


temp. 

dens. 

0 
2 
4 
6 

8 
10 

.99984 
.99994 
.99997 
.99994 
.99985 
.99970 

12 

14 
16 

18 
20 

.99950 
.99924 
.99894 
.99859 
.99820 

22 
24 
26 

28 
30 

.99777 
.99730 
.99678 
.99623 
.99564 

X 

y   ^y 

0 

0 

2 

3 

4 

12 

6 

27 

8 

47 

10 

73 

12 

103 

14 

138 

16 

177 

18 

220 

20 

267 

22 

319 

24 

374 

26 

433 

RELATIONSHIP   BETWEEN    TEM- 
PERATURE AND  DENSITY  OF  WATER. 


GRAPHIC  ANALYSIS 


135 


them  in  the  vacant  column  headed  ^y.  Since  the  x's 
in  the  first  column  and  the  ^y'&  in  the  third  one  are 
proportional  their  relative  magnitudes  can  be  determined 
by  the  black  thread  method. 

Plot  their  values  on  another  graphic  diagram,  unless 
special  directions  to  the  contrary  are  given  by  your 
instructor,  and  determine  the  slope  with  the  black  thread. 
Since  this  is  a  case  of  proportion  the  thread  must  of 
necessity  pass  through  the  origin  (§  114,  If  2),  even  though 
it  may  appear  to  lie  less  evenly  along  the  row  of  points 
than  it  otherwise  would.  (Measurements  which  must 
necessarily  fulfill  a  certain  conditional 
relationship  are  called  conditioned 
measurements  and  will  be  considered 
later.  In  this  case  notice  that  the 
temperature  and  density  of  water  are 
not  in  themselves  conditioned  meas- 
urements, but  we  are  attempting  to 
make  them  satisfy  a  "  condition," 
namely,  that  they  shall  follow  the  law 
y  =  a  +  bx  +  ex.2)  The  value  ob- 
tained for  the  slope  of  the  black  thread 
will  probably  be  in  the  neighborhood 
of  5/6  or  0.83. 

If  V  y  =  -83x,  then,  it  follows  that 
y  •=  .69z2,  y  being  expressed  in  hun- 
dred-thousandths as  in  column  2  of  the 
second  table  and  being  measured  downward  from  the 
level  of  the  vertex  of  the  parabola.  Instead  of  measuring 
^/-values  downward  from  the  original  99997  it  will  be 
found  easier  to  increase  them  by  3,  as  in  the  third  column 
of  the  next  table,  and  then  measure  the  results  down- 
ward from  the  level  100000,  which  is  probably  one  of 
the  ruled  lines  of  the  plotting  paper. 


X 

.69x2 

2/  +  3 

0 

0 

3 

2 

3 

6 

4 

11 

14 

6 

25 

28 

8 

44 

47 

10 

69 

72 

12 

99 

102 

14 

135 

138 

16 

177 

180 

18 

224 

227 

20 

276 

279 

22 

334 

337 

24 

398 

401 

26 

467 

470 

TABLE  OF  VAL- 
UES FOR  y  =  .69z2 
AND  FOR  .69rc2  +  3. 


136  THEORY  OF  MEASUREMENTS  §119 

On  the  graphic  diagram  already  used  for  plotting  the 
density  of  water  lay  off  the  values  of  y  +  3  (in  .OOOOl's 
of  a  unit  of  density  proper)  downward  from  the  line 
y  =  1.00000,  noticing  particularly  that  x  =  0  is  now  at 
the  vertex  of  the  curve  (at  4°,  not  at  0°).  If  the  work 
has  been  carefully  done  it  will  be  seen  that  this  new  curve 
of  the  equation  y  =  —  .69z2,  or  more  accurately, 
100000?/*  =  —  .69z2  forms  a  fairly  good  approximation 
to  the  unknown  relationship  of  the  empirical  values  of 
density  and  temperature. 

The  last  step  that  remains  to  be  taken  is  to  reduce  the 
equation  100000?/  =  -  .69z2  to  the  original  axes  of 
temperature  and  density.  Since  x  is  zero  when  t  (tem- 
perature) is  4,  and  2  when  t  is  6,  etc.,  it  is  plain  that 
x  =  t  —  4  every  where; ;in  the  same  way  y  =  d  —  0.99997. 
Substituting  these  values  in  the  original  equation 
lOOOOOi/  =  -  .69z2  gives 

d  =  0.99986  +  0.0000522J  -  0.0000069J2 

which  is  the  required  law  expressing  the  relationship  be- 
tween density  and  temperature. 

119.  Typical  Curves. — Any  law  of  change  (as  far  as 
finite  values  of  the  variables  are  concerned)  can  be 
expressed  by  an  equation  of  the  form  y  =  a  +  bx 
+  ex2  -f  dx*  +  ex4  +  •  •  •  if  enough  terms  are  used, 
but  it  will  easily  be  understood  that  the  method  soon 
becomes  difficult  to  handle.  Sometimes  the  appearance 
of  the  curve  makes  it  possible  to  guess  that  its  equation 
is  of  some  particular  form  (Figs.  29-34),  or  the  form 
may  be  deducible  from  theoretical  considerations  (c/. 

*  The  large  coefficient  is  needed  because  the  ^/-values  do  not 
properly  run  into  the  hundreds  of  units  as  would  appear  from  the 
table  but  are  condensed  within  a  small  fractional  range  (.99564  to 
.99997)  if  they  are  to  represent  densities  correctly.  See  §  109,  and 
§  111,  No.  4. 


GRAPHIC  ANALYSIS 


137 


Fig.  33).  For  a  curve  that  deviates  only  slightly  from  a 
straight  line  it  often  happens  that  the  black  thread 
method  will  give  a  linear  law  that  is  a  sufficiently  good 
approximation  for  practical  purposes.  The  parabola 
can  usually  be  fitted  fairly  well  to  a  curve  that  shows  a 
single  upward  or  downward  sweep,  and  is  easier  to  apply 
than  the  exponential  curve,  which  is  so  often  used  instead. 

120.  Linear  Relationship  by  Change  of  Variables.— 
In  the  previous  section  the  values  of  y  were  not  pro- 
portional to  those  of  x,  but  a  straight  line  was  obtained 
by  plotting  x  and  V  y  instead  of  x  and  y.  A  transforma- 
tion of  this  sort  can  always  be  made  when  the  type  of 
equation  has  been  picked  out  and  its  numerical  constants 
are  to  be  determined.  Thus  if  y  =  eax  the  transformed 
equation  log  y  =  ax  log  e  shows  at  once  that  x  and  log  y 
are  proportional;  if  y  =  ax/(b  -f-  x)  it  will  be  found  that 
y  and  y/x  follow  the  straight  line  law;  if  y  =  a/x  either 
of  the  variables  will  be  directly  proportional  to  the 
reciprocal  of  the  other;  etc. 

The  volume  of  a  certain  mass 
of  air  was  found  to  vary,  under 
changes  of  pressure  to  which 
it  was  subjected,  according  to 
the  numbers  in  the  following 
table.  Assuming  that  the 
pressure  and  volume  are  in- 
versely proportional,  represent 
the  relation  between  them  by 
an  equation  after  plotting  a 
smoothed  curve  on  a  graphic 
diagram  with  a  condensed  x- 
scale  (pressures)  and  an  ex- 
panded i/-scale  (volumes)  : 
Since  y  is  inversely  proportional  to  #  a  new  variable, 


pressure 

volume 

760  mm  Hg 

8.1  cm3 

830 

7.2 

926 

6.6 

1022 

5.8 

1125 

5.3 

1230 

4.9 

1340 

4.5 

1410 

4.1 

1520 

4.0 

1600 

3.9 

EXPERIMENTAL  DETER- 
MINATION OF  pv- VARIATION  OF 
A  GAS. — The  law  of  varia- 
tion for  a  constant  mass  of 
gas  is  known  to  be  of  the  form 
v  =  k/p. 


138 


THEORY  OF  MEASUREMENTS 


§120 


X  GRAPHIC  ANALYSIS  139 

l/y,  can  be  obtained,  which  will  vary  directly  with  x. 
By  plotting  x  and  l/y  and  holding  a  black  thread  so  as 
to  pass  through  the  origin  (§  114,  If  2)  find  an  equation 
connecting  x  and  l/y,  and  reduce  it  to  a  simple  equation 
between  x  and  y.  The  result  will  probably  be  in  the 
neighborhood  of  v  =  6000/p.  Plot  this  equation  on  the 
same  graphic  diagram  as  the  smoothed  curve  and  notice 
how  closely  the  two  loci  correspond. 

FIG.  29.  THE. STRAIGHT  LINE. — Its  equation  may  be  written 
either  y  =  a  =fc  (a/m)x,  taking  care  that  the  right  sign  is  used  for 
the  gradient;  or  x/m  +  y/a  =  1,  if  the  signs  of  m  and  a  are  taken 
according  to  the  usual  convention. 

FIG.  30.  THE  PARABOLA  ay  =  x2. — The  equation  shows  that  it 
cuts  the  line  y  =  x  at  the  point  (a,  a).  While  running  infinitely 
far  upward  the  curve  also  extends  infinitely  far  to  the  right  or  left, 
but  approaches  verticality,  and  its  ends  subtend  an  angular  distance 
that,  seen  from  the  vertex,  approaches  zero.  All  parabolas  are 
similar  figures. 

FIG.  31.  THE  RECTANGULAR  HYPERBOLA  xy  =  a2. — The  curve 
consists  of  two  separate  parts  (" branches"),  each  of  which  extends 
to  an  infinite  distance  and  approaches  two  fixed  lines,  called  its 
asymptotes,  without  ever  reaching  them.  All  rectangular  hyperbolas 
are  similar  figures. 

FIG.  32.  THE  CURVE  y  =  a*fx2. — I'js  equation  shows  that  it 
passes  through  the  points  (=fc  a,  +  a),  (±  0,  +  QO  ),  and  (db  °°,  +0), 
and  that  it  is  symmetrical  with  respect  to  the  ?/-axis  since  there  is  a 
point  (—  m,  +  n)  corresponding  to  every  point  (+  m,  +  n)> 

FIG.  33.  THE  EXPONENTIAL  CURVE  y  =  e*. — A  remarkable 
property  of  this  curve  is  that  its  slope  is  everywhere  equal  to  its 
ordinate  at  the  corresponding  point.  In  the  more  general  equation 
y  =  emx  the  slope  is  proportional  to  the  ordinate,  so  that  the  curve 
may  be  used  to  represent  a  relationship  like  Newton's  Law  of  Cooling; 
viz.,  the  rate  at  which  a  body  loses  temperature  is  proportional  to 
the  temperature  itself.  Curves  for  m  =  2,  m  =  —  1  and  m  =  —  1/2 
are  shown  by  dotted  lines. 

FIG.  34.  THE  CURVE  OF  ERRORS  yfb  =  e~(xfa)2. — Three  different 
values  of  a  are  represented  for  a  large  value  of  b  and  one  of  a  for  a 
smaller  one  of  6. 


140  THEORY  OF  MEASUREMENTS  §121 

121.  Questions  and  Exercises.  —  1.  With  the  black 
thread  find  the  best  straight-line  approximation  for 
v  =  6000/p  or  for  your  smoothed  curve.  Ans.  :  approxi- 
mately y  =  7.7  —  A7x. 

2.  Name  the  kind  of  curve  that  would  correspond  to 
the  equation  obtained  by  solving 

8.1  =  a  +  6(760)  +  c(760)2, 
5.3  =  a  +  6(1125)  +  c(1125)2, 
4.0  =  a  +  6(1520)  +  c(1520)2, 

for  a,  6,  and  c,  and  substituting  the  values  so  obtained 
in  the  equation  y  =  a  +  bx  +  ex2.  For  what  purpose 
could  the  resultant  equation  be  used? 

3.  Explain    how    an    equation    of   the    form    y  —  eax 
could  be  determined  for  the  smoothed  curve  of  the  last 
section. 

4.  Show  that  the  distance  between  the  points  (3,  4) 
and  (7,  5)  is  equal  to  i/(3  -  7)2  +  (4  -  5)2,  and  write 
a  general  formula  for  finding  the  distance  between  any 
two  points,  such  as  (zi,  yj  and  (x2,  2/2). 

5.  The  equation  of  the  straight  line  that  passes  through 
the  two  points  (xi,  y\)  and  (z2,  2/2)  is 

2/2  - 


Draw  a  diagram  of  the  line  and  the  two  points,  and 
explain  the  significance  of  the  subtracting,  the  dividing, 
and  the  equating  in  the  above  formula. 

6.  Guess  at  the  equation,  of  table  9  in  §101.     Test 
the  equation  by  substituting  a  few  tabular  values,  and 
if  your  first  estimation  was  faulty,  make  a  better  one. 

7.  Prove  that  every  point  on  the  parabola  of  Fig.  30 
is  as  far  from  the  line  y  =  —  a/4  as  it  is  from  the  point 


X  GRAPHIC  ANALYSIS  141 

(0,  +  a/4).     The  line  is  called  its  directrix  and  the  point 
its  focus. 

8.  If  eax  is  always  identical  with  (ea)x  and  if  2.7182-303 
is  equal  to  10,  turn  to  Fig.  33  and  see  how  the  curve 
y  =  10*  would  run.  (The  reciprocal  of  2.303  is  .4343.) 
Decide  how  the  curve  x  =  I0y  would  compare  with  it. 
Have  you  drawn  the  latter  curve  previously? 


XI.  INTERPOLATION  AND  EXTRAPOLATION 

Apparatus. — Black  thread;  slide  rule;  pencil  with  a 
sharp  point. 

122.  Definitions. — The  process  of  drawing  the  locus 
of  an  equation  by  plotting  a  few  isolated  points  and  filling 
up  the  intermediate  positions  with  a  smooth  curved  line 
involves  the  tacit  assumption  that  the  values  of  y  for  the 
intervening  values  of  x  would  have  been  found  to  vary 
in  this  predeterminate  manner  if  they  had  been  calculated. 
That  is,  the  value  of  a  function  that  corresponds  to  a 
certain  magnitude  of  its  independent  variable  need  not 
be  obtained  by  calculation  in  all  cases,  but  may  often  be 
determined  by  comparing  it  with  the  values  which  the 
function  is  known  to  have  when  the  variable  is  larger 
and  smaller  than  in  the  particular  case  that  is  under  in- 
vestigation. When  a  few  known  values  are  used  for  the 
purpose  of  determining  an  intermediate  unknown  value 
the  latter  Is  said  to  be  found  by  interpolation.  For 
example  if  the  population  of  a  city  were  known  for  each 
of  the  years  1850,  1860,  1870,  1880,  1890,  and  1900,  one 
could  guess  fairly  accurately  what  the  population 
amounted  to  in  the  year  1875,  even  if  the  given  values 
should  not  follow  any  known  law  or  any  recognizable 
type  of  curve. 

The  process  of  using  a  certain  range  of  values  for  de- 
termining a  value  that  lies  outside  of  that  range  is  called 
extrapolation  (Latin,  extra,  outside;  inter,  between; 
polire,  to  make  smooth) .  Thus,  from  the  data  mentioned 
above  it  would  be  possible  to  make  some  kind  of  an 
estimate  of  the  population  for  the  year  1840,  or  for 
1910,  or  perhaps  even  for  1920. 

142 


XI  INTERPOLATION  AND  EXTRAPOLATION          143 

Turn  to  your  diagram  of  the  daily  variations  in  body 
temperature  (§  99)  and  determine  the  normal  temper- 
ature of  the  human  body  at  9:30  A.M.  It  should  be 
about  37°.07,  .08,  or  .09.  What  was  the  temperature  of 
this  individual  at  9:30  A.M.  on  the  day  of  the  experiment. 
Ans.:  probably  about  37°.15  or  37°.16. 

Interpolation  is  a  process  that  is  trustworthy  only 
when  the  data  are  sufficiently  numerous  and  are  given 
at  sufficiently  close  intervals,  and  when  their  variation 
is  not  too  irregular.  It  would  be  impossible  to  inter- 
polate the  values  y  =  x*  —  x  (Fig.  25)  from  the  three 
points  (-  1,0),  (0,  0),  (  +  1,  0);  or  to  fill  in  a  free-hand 
parabola  for  y  =  x2  —  2x  —  3  if  the  only  data  were  the 
points  (-  1,  0)  and  (+2,  -  3). 

123.  The  Principle  of  Proportionate  Changes.  —  It  is 
always  necessary  to  make  an  assumption 
of  some  kind  when  a  process  of  interpo- 
lation is  used.  In  obtaining  logarithms 
from  a  table  by  interpolation  it  is  as- 
sumed, for  example,  that  the  logarithm  ELEMENTARY 
of  2718  is  8  tenths  of  the  way  from  log  STRAIGHTNESS. 
2710  to  log  2720  (§66),  or  in  general  -When  a  circle 
that  any  change  in  a  logarithm  is  pro-  that  is  drawn 
portion  al  to  the  change  in  the  corre-  around 


spending  natural  number.     This  is  an  °n     a 

smooth  curve   is 
instance  of  the   linear  law  (§114),  and     made    smaller 

may  be  expressed  as  A  (log  x)  =  k&x.  and  smaller  it  is 
It  is  true  only  for  small  differences  (log  cut  more  and 
300  -  log  200  is  not  equal  to  log  200  more  nearly  into 
-  log  100),  because  thecurveof,  =  log 
x  is  nowhere  nearly  a  straight  line  unless 
a  very  small  stretch  of  it  is  considered  by  itself. 

Plot  a  graphic  diagram  of  y  =  log  x  from  x  =  0  to 
x  =  10  using  a  large  scale  on  the  z-axis  (5  n's  =  1  unit) 


144  THEORY  OF  MEASUREMENTS  §123 

and  a  much  larger  one  on  the  ?/-axis  (20  n's  =  1  unit). 
Do  not  use  any  z-values  except  0.1,  0.2,  0.4,  0.6,  1.0, 
2.0,  3.0,  •  •  •  10.0.  Draw  a  free-hand  curve  as  smoothly 
as  possible  through  the  corresponding  points.  Measure 
the  ^/-values  for  x  =  3.5,  2.5,  1.4,  and  0.8;  then  verify 
each  by  referring  to  the  tables.  Notice  that  the  points 
on  the  curve  for  which  x  =  1,  2,  3,  do  not  lie  in  a  straight 
line.  Notice  that  the  short  stretch  of  curve  that  includes 
the  points  x  =  2710,  2718,  2720  has  no  perceptible  cur- 
vature. If  the  points  are  specified  with  sufficient  ac- 
curacy, however,  it  will  be  found  that  no  three  of  them 
lie  in  the  same  straight  line;  thus  four-place  logarithms 
may  be  safely  interpolated  if  the  values  for  every  three- 
figure  natural  number  are  given,  but  five-place  accuracy 
for  the  logarithms  necessitates  four-figure  values  for  the 
natural  numbers  in  the  first  part  of  a  logarithm  table, 
that  corresponds  to  the  more  sharply  curved  part  of  the 
graph  (compare  tables  in  appendix). 

The  fact  that  a  curve  is  " smooth"  means  that  a  very 
short  stretch  of  any  part  of  it  deviates  very  little  from 
a  straight  line.  In  other  words,  if  a  small  circle  is  de- 
scribed around  any  point  of  such  a  curve  its  circumference 
will  be  cut  by  the  curve  at  two  points  whose  angular 
distance  approaches  180°  as  the  circle  is  made  smaller 
and  smaller  (Fig.  35).  This  property  of  a  curve  is  known 
as  elementary  straightness,  the  term  " element"  being 
used  in  the  sense  of  "a  very  small  portion,"  and  is  char- 
acteristic of  the  curves  of  all  equations  in  which  y  can 
be  expressed  as  a  rational  function  of  x.  Some  ."  tran- 
scendental" equations*  lack  elementary  straightness  at  a 
few  points,  but  in  general  the  process  of  linear  inter- 

*  A  transcendental  equation  is  one  that  involves  non-algebraical 
functions;  for  example,  y  =  a;  log  a:.  The  curve  of  this  equation 
cuts  the  elementary  circle  around  (0,  0)  at  only  one  point,  i.  e., 
comes  to  an  abrupt  end  at  the  origin. 


XI 


INTERPOLATION  AND  EXTRAPOLATION 


145 


polation  is  applicable  to  any  tabular  values  that  are 
given  at  sufficiently  small  intervals. 

124.  Examples  of  Linear  Interpolation. — Consult  the 
table  in  §  118  and  state  the  density  of  water  at  21°  C. 

Water  boils  at  100°  C.  when  the  barometric  pressure 
is  equal  to  that  of  a  760-mm.  column  of  mercury;  to 
make  it  boil  at  101°  C.  the  pressure  must  be  raised  to 
788  mm.  Hg.  What  should  an  accurate  thermometer 
register  in  boiling  water  when  the  barometer  stands  at 
775? 

Plot  a  few  points  for  the  equation  y  =  x2  —  2  and  fill 
out  a  free-hand  curve.  When  x  is  1  y  is  —  1;  when  x 
is  2  y  is  -f-  2.  Accordingly,  if  the  locus  were  not  curved 
it  would  intersect  the  z-axis  at 
x  =  1J.  Substituting  x  =  1.3 
gives  y  =  —  .31;  substituting 
x  =  1.4  gives  —  .04  ;  substi- 
tuting x  =  1.5  gives  +  .25. 
The  stretch  of  the  curve  that 
lies  between  1.4  and  1.5  (Fig. 
36)  is  so  nearly  straight  that 
the  point  where  it  intersects 
the  #-axis  can  be  found  quite 
accurately  by  the  law  of  pro- 
portionality :  Ay  =  .29  for 
Ax  =  0.1,  so  Ay  should  equal 
the  .04  required  to  bring  y 
up  to  zero  for  .04/.29  of  0.1, 
or  .014,  beyond  1.4.  Sub- 


FIG.  36.  SOLUTION  BY 
"DOUBLE  POSITION." — Any 
equation  can  be  solved  ap- 
proximately by  aid  of  a 
graph.  Then  by  using  two 
z-values  that  are  near  the  re- 
quired root,  x,  and  interpolat- 
ing a  y-value  along  a  straight 
line  it  is  possible  to  find  as 
close  an  approximation,  P,  as 
is  desired. 


stituting  x  =  1.414  gives 
y  =  —  .000604;  substituting  x  =  1.415  will  be  found  to 
give  y  =  +  .002225.  Another  application  of  the  prin- 
ciple of  proportionality  gives  x  =  1.414  +  .000214  or 
1.414214,  and  thus  the  calculation  of  the  value  of  V2 
11 


146 


THEORY  OF  MEASUREMENTS 


§124 


proceeds  with  continually  increasing  rapidity.  This,  of 
course,  is  a  determination  of  the  value  of  one  root  of  the 
equation  0  =  x2  —  2,  and  is  known  as  the  method  of 
false  position,  or  of  double  position.  It  can  be  used  for 
any  equation  whatever  containing  one  unknown  quan- 
tity by  arranging  it  in  the  form  0(z)  =  0*  and  drawing 
a  graphic  diagram  of  y  =  <l>(x). 

A  delicate  beam  balance  has  a  pointer  which  swings  to 
a  position  of  rest  at  8.0  on  an  arbitrary  scale  when  an 
unknown  mass  is  balanced  against  standard  weights  of 
14.837  grams  but  stops  at  10.5  when  the  weights  are 
changed  to  14.836.  What  weights  would  be  required 
to  bring  the  pointer  to  the  central  position,  which  is  at 
10.0  on  the  scale?  Ans.:  14.837  -  f(.OOl);  or  14.8362. 

A  telephone  company  charges  for  measured  service  at 
the  yearly  rates  given  in  the  table.  It  will  be  plain  that 
an  increase  of  200  messages  means  an 
additional  cost  of  nine  dollars,  making 
the  rate  for  an  increased  number  of 
messages  equal  to  $9/200  m,  or  4.5c. 
per  message.  At  this  rate  600  mes- 
sages should  cost  27  dollars  ;  accord- 
ingly, it  is  evident  that  the  rate  for 
any  number  of  messages  is  made  up 
of  two  parts,  a  flat  charge  of  $21  plus 
a  message  rate  of  $9  per  hundred.  If 
the  number  of  messages  is  represented 
by  x  and  the  total  cost  by  y,  then  the 
equation  connecting  them  will  be  y  =  21  -f  .045:r,  show- 
ing straight-line  variation  without  proportionality. 

*  The  expression  $(rc)  means  "any  function  of  x,"  and  may  be 
used  as  a  general  form  to  denote  3z2,  log  x,  2ax,  or  any  other  function^ 
just  as  the  general  symbol  m  may  be  used  to  stand  for  the  number 
2,  or  100,  or  any  number  whatever.  As  alternative  notations 
/(  ),  and  F.(  )  are  often  used. 


no.  of 
messages 


charge 


600 

800 

1000 

1200 


$48 
57 
66 
75 


EXAMPLE  OF  A 
"READ  INESS-TO- 
SERVE"  CHARGE 
INCORPORATED 
WITH  A  RATE 
CHARGE. 


XI  INTERPOLATION  AND  EXTRAPOLATION          147 

125.  Graphic  Interpolation. — In  general,  tabular  values 
do  not  follow  the  straight-line  type  of  variation,  and 
rather  complicated  formulae  may  be  required  for  purposes 
of  interpolation.     In  case  a  graph  can  be  drawn,  how- 
ever,  there  is  usually  no   difficulty  in  constructing  a 
smooth  curve  (or  " smoothed"  curve,  as  occasion  may 
require)  and  obtaining  any  intermediate  values  simply 
by  measuring  them  on  the  graph.     The  process  is  some- 
times uncertain  or  erroneous  if  the  given  values  are  not 
close  enough  together  or  if  their  variation  is  too  irregular. 
It  will  have  been  noticed  that  the  problem  of  finding 
intermediate  values  is  closely  allied  to  the  problem  of 
finding  a  law  of  variation  or  of  finding  the  equation  of  a 
given  curve.     In  case  a  law  or  equation  is  known,  unless 
it  is  a  complicated  one  it  will  usually  be  found  easier  to 
substitute  and  calculate  values  than  to  interpolate  them. 

126.  Graphic  Tables. — If  the  variation  of  two  quanti- 
ties is  known  to  -follow  the  linear  law  it  is  often  conveni- 
ent to  make  a  graphic  table  by  plotting  any  two  points 
and  ruling  a  straight  line  through  them. 

Lay  off  a  scale  of  values  from  0  to  20  along  the  z-axis 
and  label  it  " inches."  Lay  off  a  scale  from  0  to  50  along 
the  ?/-axis  and  label  it  "  centimetres."  Rule  a  straight 
line  through  the  two  points  (0,  0)  and  (13,  33).  Explain 
how  a  diagram  of  this  sort  can  be  utilized. 

According  to  Hooke's  Law,  the  difference  in  length 
(stretching)  of  a  spring  is  proportional  to  the  difference 
in  force  applied  to  it.  If  a  spring  that  is  hung  in  front 
of  a  scale  has  a  length  of  12  cm.  when  no  weight  is 
attached  to  it,  and  becomes  14.85  cm.  long  when  a 
weight  of  1  gm.  is  hung  on  it,  construct  a  graphic  table 
which  will  enable  you  to  reduce  its  indicated  centimetres 
to  grams  of  weight  (see  §  104). 

127.  Interpolation  Along  a  Curve. — If  the   change   in 


148  THEORY  OF  MEASUREMENTS  §128 

two  variables  is  not  in  accordance  with  a  linear  law  it  is 
possible  to  use  certain  interpolation  formulae  for  obtaining 
intermediate  values,  but  it  is  usually  much  easier  to 
make  use  of  graphic  methods.  The  known  data  are 
plotted  as  a  series  of  points,  and  these  are  either  con- 
nected by  a  smooth  curve  or  are  investigated  with  a  view 
to  discovering  an  equation  that  will  adequately  represent 
them.  If  they  appear  to  lie  along  a  curve  that  has  a 
vertical  or  a  horizontal  asymptote  (Fig.  31)  the  hyper- 
bola xy  =  k  may  be  tried,  with  a  suitable  choice  of  tem- 
porary axes  and  scales.  If  the  curve  has  a  single  upward 
or  downward  sweep  the  exponential  curve  y  =  eax  is 
frequently  used,  but  the  parabola  y  =  ax2  is  generally 
easier  to  handle  and  can  usually  be  fitted  to  the  given 
points  just  as  satisfactorily.  It  may  be  turned  so  as  to 
have  its  axis  horizontal,  if  this  position  seems  more  suit- 
able, by  interchanging  the  variables  and  writing  the 
equation  ax  =  y2. 

In  trying  to  fit  a  parabola  to  the  part  of  the  curve  of 
y  =  log  x  that  lies  between  x  =  5  and  x  =  15  would  you 
prefer  to  have  its  axis  horizontal  or  vertical?  If  vertical, 
would  its  vertex  be  directed  upward  or  downward?  If 
horizontal,  would  its  vertex  be  directed  to  the  left  or  to 
the  right?  Plot  a  logarithmic  curve  rapidly,  on  a  small 
scale,  if  there  is  any  difficulty  in  answering  the  questions; 
compare  it  with  the  curves  for  y  =  x2,  y  =  —  x2,  x  =  y2, 
and  x  =  —  y2. 

128.  Insufficiency  of  Data. — When  certain  tabular 
values  are  given  and  others  are  to  be  obtained  by  inter- 
polation it  must  always  be  remembered  that  the  known 
values  are  the  only  actual  data  and  that  nothing  else  can 
be  obtained  without  making  some  kind  of  an  assump- 
tion (§  123).  If  it  is  assumed  that  the  points  all  lie 
along  a  smooth  curve  there  is  always  a  possibility  that 


XI  INTERPOLATION  AND  EXTRAPOLATION          149 

the  assumption  is  incorrect.  It  may  even  happen  that 
the  given  points  appear  to  be  irreconcilable  with  a 
smooth  curve  or  with  a  uniform  law,  as  in  the  following 
case :  Electricity  is  sold  by  the  kilowatt-hour  (abbreviated 
KWH.)  and  four  consumers  pay  the  same  rate  to  one 
company.  The  first  is  charged  $1.08  for  9  KWH.;  the 
second,  $1.47  for  21  KWH.;  the  third,  $0.99  for  11  KWH.; 
and  the  fourth,  $1.62  for  18  KWH.  Find  the  rate. 

Plot  the  four  points,  using  1  square  along  the  z-axis 
for  each  KWH.  and  1  square  along  the  ?/-axis  for  each 
$0.10.  It  will  be  seen  that  it  is  impossible  to  decide 
where  a  smooth  curve  should  run.  This  is  a  case  of 
what  is  called  a  "step  meter  rate"  and  gives  a  broken 
line,  not  a  smooth  curve.  The  rate  is  "  12c.  per  KWH. 
if  less  than  10  KWH.  are  used;  9c.  per  KWH.  if  the  con- 
sumption is  between  10  and  15  KWH.;  7c./KWH.  if 
over  15."  Draw  the  locus  for  this  rate  on  the  same 
graph  and  notice  that  it  passes  through  the  four  points. 
The  objectionable  feature  of  sometimes  charging  less 
when  the  use  of  the  current  is  greater  is  so  apparent  that 
a  rate  of  this  kind  is  not  often  used.  The  following  is  a 
"block  meter  rate"  which  is  not  so  objectionable  and 
gives  approximately  the  same  income  to  the  company: 
"lOc.  each  for  the  first  10  KWH.  used,  7c.  each  for  the 
next  5,  3c.  each  for  all  after  the  15th."  Plot  this  rate  on 
the  same  diagram;  also  the  rate,  "8c.  per  KWH.,  with  a 
minimum  charge  of  50c."  Find  a  smooth  curve  which 
will  come  fairly  close  to  all  three  of  these  rates.*  (Sug- 

*  In  general,  a  "broken  line"  cannot  be  represented,  by  itself, 
without  using  an  equation  containing  an  infinite  series.  This  is 
the  case  with  the  last  of  the  above  rates,  which  is  represented  by  the 
straight  line  y  =  50  from  x  =  0  to  x  =  6.25  only,  and  the  straight 
line  y  =  Sx  for  the  part  further  to  the  right  only. 

If  we  are  not  restricted  to  finite  stretches  of  lines  it  is  always 
possible  to  find  an  equation  for  both  of  two  loci  each  of  which  has 


150 


THEORY  OF  MEASUREMENTS 


§129 


0^ 


gestion:  consider  the  equation  y  =  I2x  —  (l/a)x2  with 
a  suitable  value  for  a.) 

129.  Use  of  Logarithmic  Paper. — A  very  common  type 
of  variation  is  that  in  which  one  of  the  variables  is  pro- 

!,!,,,  t  ...  f     Portional  to  some  power  of 

the  other  one.  For  ex- 
ample, the  distance  trav- 
ersed by  a  falling  body  is 
proportional  to  the  square 
of  the  time  of  fall  ;  the 
time  of  rotation  of  a  planet 
or  satellite  about  a  particu- 
lar central  body  is  propor- 
tional to  the  3/2  power  of 
its  mean  distance;  friction 
of  water  flowing  through  a 
pipe  varies  (approximately) 
as  the  1.8  power  of  the  ve- 
locity. The  determination, 
for  a  set  of  experimental 
data,  of  the  proper  value 
of  the  exponent  is  not  easily  accomplished  by  means  of 
an  ordinary  graph,  because  the  various  curves  y  =  x2, 
y  =  x3,  y  =  x*,  etc.,  all  have  the  same  general  shape 
(Fig.  37).*  If  logarithms  are  taken,  however,  of  both 
sides  of  the  equation  y  =  xn  the  equivalent  equation 
log  y  =  n  log  x  is  obtained,  showing  that  log  x  and  log  y 

its  own  ascertainable  equation,  by  putting  the  individual  equations 
in  the  form  </>(z,  y)  =  0  and  multiplying  them  together.  Thus, 
the  two  lines  y  =  50  and  y  =  8x  are  the  locus  of  the  single  equation 
(y  -  50)  X  (y  -  8z)  =  0,  or  400z  -  Sxy  -  5Qy  +  y2  =  0,  as  the 
student  can  readily  show  by  plotting  a  few  points  or  by  algebraical 
treatment  (compare  §  111;  8,  d.). 

*  Experimental  measurements  are  usually  of  positive  quantities, 
so  that  the  shape  of  these  curves  to  the  left  of  the  i/-axis  or  below 
the  x-axis  is  not  a  determining  factor. 


FIG.  37.  Loci  OF  y  =  xn. — 
The  curves  are  drawn  for  n  = 
1,  2,  3,  4,  10,  50,  and  100. 


XI          INTERPOLATION  AND  EXTRAPOLATION          151 


10 


are  proportional.  Accordingly,  if  log  x  and  log  y  are 
plotted  on  an  ordinary  graphic  diagram  a  straight  line 
through  the  origin  will  be  obtained.  Now,  just  as  a 


2  3  4          5       6       7     8     9    10 

FIG.  38.  LOGARITHMIC  PAPER. — The  logarithmic  scales  cause 
the  graph  of  any  equation  of  the  form  y  =  axn  to  appear  like 
y  =  a  +  nx  on  ordinary  paper,  viz.,  as  a  straight  line. 

slide  rule  is  constructed  by  marking  the  numbers  1,  2, 
3,  4,  etc.,  at  distances  which  are  really  log  1,  log  2,  log  3, 


152  THEORY  OF  MEASUREMENTS  §129 

log  4,  etc.,  so  plotting  paper  can  be  constructed  with 
logarithmic  scales  like  those  of  the  slide  rule  along  each 
axis;  consequently  plotting  x  and  y  according  to  the 
numbered  scales  will  really  be  plotting  points  at  actual 
distances  of  log  x  and  log  y  from  the  origin;  and  if  these 
logarithms  are  known  to  be  proportional  it  will  be 
evident  that  the  graph  of  y  =  xn  will  be  a  straight  line 
through  the  origin.  Such  paper  is  called  logarithmic  paper 
and  can  be  obtained  from  dealers  who  handle  drawing 
materials.  The  lower  left-hand  corner  of  the  sheet  is  the 
origin  and  is  marked  1,1,  meaning  of  course  log  1,  log  1, 
or  0,  0;  and  the  scales  extend  both  upward  and  to  the 
right  from  1  to  10  as  in  the  C  and  D  scales  of  the  slide 
rule,  or  sometimes  from  1  to  100  as  in  the  A  and  B  scales. 
The  latter  arrangement  would  allow  a  single  unbroken 
line  to  be  used  to  represent  the  "curve"  of  y  =  V# 
(Fig.  38),  although  of  course  such  a  line  could  not  be 
drawn  for  the  complete  locus  of  y  =  x3. 

If  a  straight  line  is  drawn  on  logarithmic  paper  without 
passing  through  the  origin  it  must  cut  the  y-&xis  at  some 
point,  such  as  k  on-  the  logarithmic  ?/-scale.  Then 
(log  y)  =  (log  k)  +  n(log  x)  (compare  the  ordinary  equa- 
tion of  the  straight  line  y  =  a  -J-  bx,  §§  113,  105),  or 
log  y  =  log  k  +  log  (xn),  or  log  y  =  log  (k  X  xn),  or 
y  =  kxn.  In  other  words,  a  straight  line  on  logarithmic 
paper  represents  the  equation  y  =  axn,  a  being  the  y- 
intercept  as  measured  by  the  logarithmic  scale,  and  n 
the  true  slope  as  measured  by  uniform  scales.  The 
diagram  (Fig.  38)  shows  y  =  V  x  and  y  =  x°  on  logarith- 
mic paper.  Any  data  that  are  suspected  of  following 
the  law  y  =  axb  may  be  plotted  directly  on  this  ruled 
diagram  and  their  equation  determined  at  once. 

130.  Semi-Logarithmic  Paper. — Consider  the  equation 
of  a  straight  line  on  paper  that  has  a  uniform  scale 


XI  INTERPOLATION  AND  EXTRAPOLATION          153 

along    the   x-axis   and  a  logarithmic   scale    along    the 


Instead  of  y  =  a  +  bx  or  log  y  =  log  a  +  &  log  x  its 
equation  must  now  be  log  y  =  log  a  +  6z.  Clearing  this 
of  logarithms  gives  y  =  a  X  10&*  or  y  =  a  X  ebx  accord- 


1          2          3          4          5          6  /  9 

FIG.  39.  SEMI-LOGARITHMIC  PAPER.— The  logarithmic  scale 
along  the  ?/-axis  causes  a  straight  line  through  the  origin  to  represent 
proportionality  between  x  and  log  y,  i.  e.,  log  y  =  kx.  The  exponen- 
tial law  of  variation,  y  =  aemx,  is  obviously  reducible  to  the  form 
log  y  =  kx. 


10 


154 


THEORY  OF  MEASUREMENTS 


§130 


ing  to  the  base  that  is  used.*  Consequently,  a  straight 
line  on  semi-logarithmic  paper  (Fig.  39)  represents  the 
(( exponential"  type  of  variation,  y  =  aemx. 

Draw  a  straight  line  or  hold  a  black  thread  on  either 
Fig.  38  or  Fig.  39  so  as  to  represent  the  area  of  a  circle 
(on  the  vertical  scale)  that  corresponds  to  the  radius 
(as  indicated  on  the  horizontal  scale).  The  formula  is 
a  =  irr2,  or  y  =  irx2. 

Plot  the  locus  of  y  =  2x  on  squared  paper  for  positive 
integral  values  of  #  up  to  6  or  7.  Plot  the  same  equation 
up  to  z=10  on  logarithmic  paper,  f  and  also  on  semi-log- 
arithmic paper.  .  In  which  case  is  the  locus  a  straight 
line?  Plot  y  =  2~x  on  semi-logarithmic  paper. 


FIG.  40.     EXTRAPOLATION  DIAGRAM. — Graph  of  the  relation  be- 
tween date  and  population. 

Explain  how  to  use  one  of  these  diagrams  (Figs.  38 
and  39)  to  make  a  graphic  table  of  the  relationship 
between  the  period  (p)  of  vibration  time  of  a  pendulum 
and  the  length  (I)  of  the  pendulum^  if  they  are  related 
according  to  the  formula  p  =  2iri/l/g,  g  being  a  constant. 

*  These  two  equations  are  of  the  same  form;  the  first  can  be 
put  into  the  form  of  the  second  merely  by  changing  the  value  of  6; 
for  10  =  e2-30,  so  10te  is  identical  with  e2-306*. 

f  If  the  specially  ruled  paper  is  not  on  hand  the  work  can  be  done 
on  thin  paper  laid  over  Fig.  38  and  Fig.  39  of  this  book. 


XI  INTERPOLATION  AND  EXTRAPOLATION          155 

On  Fig.  38  or  Fig.  39  indicate  the  straight  line  that 
represents  the  equation  xy  =  a  (suggestion:  y  =  ax~l). 

131.  Extrapolation. — The  principles  of  extrapolation 
are  like  those  of  interpolation,  but  the  former  process  is 
naturally  more  uncertain  than  the  latter  and  can  be 
trusted  to  give  good  results  only  when 


the   extrapolated  point   is   relatively  year 
near   the   points  that  correspond    to 

the  known  data.     As  an  example  of  i860 

the  use  of  extrapolation  the  table  and  }|70 


1850 


1880 


population 


93,000 
380,000 
560,000 
865,000 
1,208,000 
1,485,000 


graph  give  the  population  of  the  state       1890 

of  California  from  1850  to  1900.     It       jjjOO 

is  required  to  find  the  population  in 

1910.     Continue  the  curve  in  the  way 

that  you  think  it  would  be  apt  to 

run,  and  note  where  it  cuts  the  1910  ordinate. 

Reconstruct  the  extrapolated  part  of  the  curve,  if 
necessary,  so  as  to  make  it  give  2.38  X  106  for  1910  and 
extrapolate  again  to  determine  its  height  for  1920. 
What  do  you  find  the  population  will  be  for  this  date? 

The  following  example  shows  that  extrapolation  may 
be  a  very  definite  and  decisive  process  if  the  given  values 
follow  a  consistent  law  of  change  and  can  be  carried 
close  to  the  required  value:  Find  the  instantaneous 
velocity  of  a  body  at  a  certain  point  of  time  if  it  is  known 
that  immediately  after  that  instant  it  travels  10  cm.  in 
the  first  second,  7.0711  cm.  in  the  first  half  second,  etc., 
as  given  in  the  next  table.  Plot  the  tabular  values  with 
a  scale  of  time  along  the  x-axis  and  a  scale  of  average 
velocity  along  the  i/-axis  and  extrapolate  graphically  to 
find  the  velocity  in  no  interval  of  time.* 

*  In  no  time  a  moving  object  will  of  course  traverse  no  space  if 
its  velocity  is  not  infinite,  and  there  is  strictly  speaking  no  meaning 
for  such  a  phrase  as  "instantaneous  velocity."  It  is  convenient, 


156 


THEORY  OF  MEASUREMENTS 


§132 


time 
(sec.) 

space 
(cm.) 

velocity 

(cm.  /sec.) 

1 
0.5 
0.3 
0.2 
0.1 
0.05 

10.000 
7.0711 
4.5399 
3.0902 
1.5643 
0.7846 

10.000 
14.142 
15.133 
15.451 
15.643 
15.692 

AVERAGE  VELOCITY  OF 
A  PISTON  ROD  THAT  MOVES 
WITH  SIMPLE  HARMONIC 
MOTION. — Its  total  path  is 
assumed  to  measure  20  cm. ; 
its  period,  4  seconds;  the 
required  velocity  is  that  at 
the  central  point  of  its 
motion. 


132.  Questions  and  Exercises. — 1.  Write  a  definition 
of  what  you  understand  by  the  terms  interpolation  and 
extrapolation. 

2.  Turn  to  the  table  of  values  for  Chauvenet's  criterion 
(§  208)  and  plot  I  as  a  function 
of  n,  using  enough  points  to  de- 
termine whether  the  values  of  I 
for  integral  values  of  n  that  are 
not  given  in  the  table  (e.  g., 
n  =  31,  n  =  70)  can  be  satisfac- 
torily obtained  by  graphic  in- 
terpolation. 

3'  Turn  to  y°ur  SraPhic  dia~ 
Sram  of  V  =  log  %  and  draw  a 
straight  line  from  the  point  (2, 
log  2)  to  the  point  (3,  log  3),  thus 
making  a  chord  of  the  curve. 
Mark  the  middle  point  of  the 
chord.  Is  the  ordinate  of  this 
point  equal  to  the  average  of  log 
2  and  log  3? 

What  operation  can  be  performed  on  any  two  numbers 
by  averaging  their  logarithms?  Label  your  diagram  so 
as  to  show  clearly  the  distance  that  corresponds  to 
log  ^  and  the  distance  that  corresponds  to  log 
[(2  X  3)1/2].  Which  of  the  5-formulse  for  approximate 
calculation  with  small  magnitudes  does  this  diagram 
illustrate?  In  what  way  does  it  show  the  equality 

however,  to  consider  that  a  body  which  speeds  up  from  a  condition 
of  rest  to  a  definite  velocity  must  have  passed  through  all  inter- 
mediate velocities  successively,  and  as  its  velocity  must  have  been 
continually  changing  the  concept  of  a  definite  velocity  at  a  certain 
point  of  space  or  time  becomes  almost  a  necessity.  For  purposes  of 
rigorously  logical  deduction  this  concept  is  defined  as  a  limiting 
value  in  the  way  indicated  above. 


XI         INTERPOLATION  AND  EXTRAPOLATION         157 

expressed  by  the  formula,  and  in  what  way  does  it  show 
that  this  equality  is  not  exact  but  only  approximate? 

4.  In  finding  the  root  of  an  equation  by  the  method 
of  double  position  would  it  be  satisfactory  to  extrapolate 
instead  of,  interpolating?     Explain  why. 

5.  Calculate  the  roots  of  the  equation 

(3.2)  (**)  =  TT  +  log  (sins). 

(Suggestions :  Draw  a  rough  diagram  of  y  =  sin  x,  say  from 
—  TT  to  +  3?r.  Then  add  a  rough  outline  of  ?/  =  log  (sin  x), 
remembering  that  log  0  =  —  <»  ,  and  that  negative  numbers 
have  no  [real]  logarithms.  Draw  also  y  =  ir,  and  then 
y  =  IT  —  (#2)  (*2).  It  will  now  be  obvious  that  the  graph  of 
y  —  [T  —  (#2)(a;2)]  +  [log  (sin  a;)]  cannot  cut  the  z-axis  in 
more  than  two  points.  Calculate  each  of  them  separately 
by  the  method  of  double  position.  Their  sum  should  be 
1.368.) 

6.  A  line  is  indicated  on  Fig.  38,  page  151,  which  passes 
through  the  points  (1,  1)  and  (3,  8)  of  the  logarithmic 
paper.     Determine  the  equation  which  it  represents: 

7.  Plot  the  locus  of  y  =  a  -f  bx  +  k/x2  and  find  the 
asymptotes  of  the  (oblique-angled)  hyperbola  which  is 
obtained. 


XII.     COORDINATES  IN  THREE  DIMENSIONS. 


Apparatus. — A  pencil  with  a  sharp  point;  a  " quad- 
rangle" or  " topographical  sheet"  of  the  U.  S.  Govern- 
ment contour  map;  (model  of  a  small  area  of  the  map, 
made  by  piling  up  contour  sections  sawn  out  of  thin 
wood). 

133.  Coordinates  of  a  Point  in  Space. — Just  as  the 
position  of  a  point  in  a  plane  (i.  e.,  in  two-dimensional 
space)  can  be  represented  by  two  coordinates,  an  z-value 

and  a  2/-value,  giving  its 
distance  from  two  mutu- 
ally perpendicular  axes, 
so  the  location  of  a  point 
in  unrestricted,  three- 
dimensional  space  can  be 
fixed  by  three  coordi- 
nates, a  set  of  three  nu- 
merical values  (a;,  y,  z) 
which  indicate  its  dis- 
tance from  each  of  three 
planes  that  intersect  each 
other  at  right  angles. 


FIG.  41.  COORDINATE  PLANES. — 
These  planes  of  reference  for  three- 
dimensional  space  correspond  to  the 
base  lines  of  reference,  or  coordinate 
axes,  that  are  used  for  two-dimen- 
sional space. 


Thus,  the  point  in  Fig. 
41  is  at  a  distance  of  x 
units  along  the  x-axis 
from  the  plane  YZ  of  the  other  two  axes,  and  at  a  dis- 
tance of  y  units,  parallel  to  the  y-axis,  from  the  plane 
XZ,  and  at  a  distance  of  z,  parallel  to  the  2-axis,  from 
the  plane  XY. 

134.  Convention  in  Regard  to  Signs. — The  XY  plane 
may  be  thought  of  as  being  represented  by  the  same  sheet 

158 


XII          COORDINATES  IN   THREE  DIMENSIONS          159 

of  paper  as  that  on  which  the  flat  graphs  of  x  and  y  have 
previously  been  traced  (Fig.  42).  Then  any  point  what- 
ever must  be  located  a  certain  distance  above  or  below 
some  definite  position  in  the  plane  of  the  paper.  The 
o>value  and  y-value  for  this  position  will  be  the  x  and  y 
of  the  point  in  space,  and  the  distance  from  the  point  to 
the  plane  will  be  the  z  of  the  point.  It  is  customary 
among  physicists  and  astronomers  to  consider  a  distance 
above  the  plane  of  the  paper  as  a  positive  value  of  z, 
and  distance  below  it  as  negative,  according  to  the 
arrangement  of  axes  shown  in  Figs.  41  and  42.  The 


X 


FIG.  42.  POSITIVE  DIRECTIONS  OF  AXES. — Z  is  considered  posi- 
tive when  measured  upward  from  the  normal  plane  of  the  x-axis 
and  7/-axis.  The  pin  shown  in  this  diagram  would  have  the  location 
of  its  head  represented  approximately  by  the  position  (5,  7,  3),  the 
numbers  in  parenthesis  being  used  for  x,  y,  and  z,  in  order. 

convention  among  pure  mathematicians  is  just  the  oppo- 
site, viz.,  distances  above  the  paper  are  called  negative 
and  those  below  are  positive  (Fig.  43).  The  distinction 
is  immaterial  in  the  greater  part  of  the  study  of  pure 
mathematics,  but  it  is  important  in  many  branches  of 
applied  science;  for  example,  the  equations  of  the  curve 
of  a  right-hand  screw  thread  in  one  system  will  represent 


160 


THEORY  OF  MEASUREMENTS 


§134 


a  reversed  or  left-hand  thread  if  the  other  system  is 
used  instead.  Accordingly,  there  is  a  tendency  among 
pure  mathematicians  to  use  the  "  physical  "  arrange- 
ment for  the  sake  of  uniformity,  and  it  is  the  only  one 
that  will  be  used  in  this  book. 

That  these  two  arrange- 
ments exhaust  the  possi- 
bilities may  be  seen  by 
taking  any  arbitrary  ar- 
rangement of  axes  at  right 
angles  to  each  other  and 
rotating  them  as  a  rigid 
figure.  They  can  always 
be  made  to  coincide  with 
either  Fig.  43,  or  Fig.  44. 


0 


FIG.  43. 


44 


FIG.  44. 


CONVENTIONS  AS  TO  SIGNS. — 
Fig.  43  corresponds  to  the  ar- 
rangement that  is  understood  in 
pure  mathematics;  Fig.  44  is  that 
of  applied  mathematics. 


The  two  are  essentially 
different  because  neither 
one  can  be  rotated  in 
three-dimensional  space  so 
as  to  coincide  with  the  other.* 

In  each  of  the  following  exercises  the  positive  direc- 
tions of  the  axes  are  indicated.  Consider  each  in  turn 
and  state  whether  it  is  like  Fig.  44  ("right")  or  the 
opposite  ("wrong"). 

*  Notice  that  a  plane  ^-diagram  drawn  with  +  Y  upward  but 
with  +  X  to  the  left  cannot  be  rotated  in  its  own  two-dimensional 
space  so  as  to  coincide  with  the  conventional  arrangement.  It  can 
be  turned  through  the  third  dimension,  however  (turning  the  upper 
surface  of  the  paper  downward),  and  made  to  coincide.  Similarly, 
a  solid  model  of  Fig.  43  would  need  to  be  turned  through  a  fourth 
dimension  of  space  before  it  could  be  made  to  coincide  with  a  solid 
model  of  Fig.  44.  Since  we  have  no  appreciation  of  a  fourth  dimen- 
sion the  two  figures  are  as  essentially  different  to  us  as  a  capital  L 
and  a  Greek  capital  r  would  seem  to  a  being  whose  sense  perceptions 
were  limited  to  space  of  two  dimensions. 


XII         COORDINATES  IN  THREE  DIMENSIONS          161 

1.  X  points  northward;    Y,  upward;   Z,  westward. 

(Ans. :  wrong.) 

2.  X,  west;  F,  up;  Z,  north.     (Ans.:  right.) 

3.  X,  down;  Y,  east;  Z,  south. 

4.  X,  down;  F,  north;  Z,  east. 

5.  X,  down;  F,  west;  Z,  north. 

6.  X,  down;  Y,  south;  Z,  east. 

7.  X,  east;   F,  north;  Z,  up. 

8.  X,  west;  F,  north;  Z,  up. 

9.  X,  west;  F,  north;  Z,  down. 

10.  X,  up;  F,  east;  Z,  north. 

135.  Loci  of  Simple  Three-Dimensional  Equations. — 
In  studying  geometrical  relationships  in  a  plane  we  have 
seen  (§  105)  that  the  equation  y  =  a  represents  all  the 
points  that  are  a  units  above  the  z-axis,  that  is,  a  straight 
line  parallel  to  the  x-axis  and  situated  at  a  distance  a 
above  it.  In  the  same  way,  the  equation  z  =  m  must 
denote  all  points  located  m  units  above  the  z?/-plane, 
e.  g.,  the  points  (2,  3,  m),  (0,  0,  m),  (0,  10,  m),  etc.,  since 
each  of  these  groups  of  values  (x  =  2,  y  =  3,  z  =  m,  for 
example)  will  satisfy  the  equation.  Similarly,  x  =  m 
or  y  =  m  will  denote  a  particular  plane  parallel  to  F0Z 
(Figs.  41  to  44)  or  to  XOZ,  and  so  perpendicular  to  the 
x-axis  or  to  the  ?/-axis,  respectively. 

Consider  next  an  equation  that  contains  only  two  of 
the  three  variables.  On  a  plane  surface  y  =  x2  is  a 
curve,  a  parabola.  In  space,  any  point  above  or  below 
any  point  of  the  curve  will  satisfy  this  equation,  for  it 
has  the  same  x  and  y,  but  a  different  z.  Accordingly,  the 
equation  represents  the  surface  that  is  made  up  of  all 
the  vertical  lines  that  can  be  passed  through  points 
that  lie  on  the  plane  curve. 

If  the  equation  contains  all  three  variables,  any  arbi- 
trary value  may  be  assigned  to  z,  any  value  whatever  to 

12 


162  THEORY  OF  MEASUREMENTS  §136 

y,  and  the  value  of  z  will  then  be  determinate.  That 
is,  the  points  of  the  locus  will  be  situated  at  varying 
distances  above  all  the  points  in  the  xy-pl&ne,  and  so 
will  comprise  a  surface. 

In  general,  then,  a  single  equation  denotes  a  surface. 
Since  two  surfaces  intersect  along  some  line  two  simul- 
taneous equations  will  denote  a  curve  in  space.  It  has 
been  seen  (§  108)  that  x2  +  y2  =  52  must  be  the  equation 
of  a  circle;  in  the  same  way  x2  +  2/2  +  z2  =  52  is  the 
equation  of  a  spherical  surface  that  is  everywhere  5 
units  distant  from  the  origin.  The  simultaneous  equa- 
tions 

*  +  y2  +  z2  =  25 
z   =    ?> 

will  have  for  their  locus  the  points  which  satisfy  the  first 
equation  and  at  the  same  time  satisfy  the  second;  i.  e., 
each  of  the  points  that  is  located  on  the  spherical  surface 
x2  +  y2  +  z2  =  25  and  is  at  the  same  time  in  the  hori- 
zontal plane  z  =  3.  Such  points  must  lie  on  the  inter- 
section of  the  plane  and  the  spherical  surface;  and  this 
intersection  is  known  to  be  a  curve,  namely,  a  circle. 
For  other  values  of  z  the  circular  intersection  would  be 
larger  or  smaller;  for  the  (tangent)  plane  z  =  5  the 
intersection  would  shrink  to  a  point,  for  z  =  6  there 
would  be  no  intersection. 

136.  Contour  Lines. — If  a  were  given  all  possible 
values  in  the  equation  z  =  a  the  resultant  loci  would  be 
horizontal  planes  at  all  levels,  and  they  would  intersect 
the  sphere  x2  +  y2  +  z2  =  52  in  all  the  horizontal  circles 
that  could  be  drawn  on  its  surface.  All  of  these  inter- 
sections, consequently,  would  be  the  spherical  surface; 
a  smaller  number  of  them  would  form  a  sort  of  skeleton, 
from  which  the  shape  of  the  surface  could  be  inferred 


XII 


COORDINATES  IN   THREE  DIMENSIONS 


163 


if  they  were  sufficiently  numerous.  An  extremely  useful 
way  of  representing  a  surface,  especially  an  irregular 
surface,  on  paper  is  by  outlining  the  intersections  which 
would  be  formed  by  horizontal  planes  at  various  levels. 
The  diagram  (Fig.  45)  shows  a  portion  of  the  surface 


H 


FIG.  45.  THE  HYPERBOLIC  PARABOLOID  x2  —  if  +  z  =  0. — A 
surface  of  this  general  character  is  sometimes  spoken  of  as  a  "saddle- 
back." 

£2  —  y2  +  z  =  0,  called  a  hyperbolic  paraboloid.  When 
x  is  zero  the  equation  reduces  to  z  =  y2',  i.  e.}  the  inter- 
section of  the  curve  with  the  FZ-plane  is  the  parabola 
z  =  y2,  BOE  in  the  diagram.  When  y  =  0  the  section 
MOJ  is  a  parabola  z  =  —  x2.  At  any  level  where  z  =  a 
the  equation  of  the  surface  becomes  y2  —  x2  =  a,  a 
hyperbola  such  as  ABC  and  DEF  or  HJK  and  LMN; 
for  a  =  0  this  degenerates  into  the  two  straight  lines 
that  are  common  asymptotes,  y  =  ±  x.  A  series  of 
horizontal  sections  of  the  surface  are  shown  in  Fig.  46 
as  they  would  appear  if  they  were  all  viewed  from  above, 


164 


THEORY  OF  MEASUREMENTS 


§137 


the   algebraical   signs   showing   whether   each   curve  is 
above  or  below  the  xy-plane  and  the  smaller  numbers 
denoting  the  lower  levels  while  the  higher  numbers  indi- 
cate the  upper  ones.     If 
y  the  numbered  lines  are 

imagined  to  be  raised  1, 
2,  3,  4,  5,  6,  7,  8,  and  9 
centimetres  respectively 
above  the  surface  of  the 
paper  it  will  be  evident 
that  a  good  idea  of  the 
shape  of  the  curved  sur- 
face which  they  outline 
can  be  obtained  without 
the  necessity  of  consult- 
ing a  perspective  drawing 
like  Fig.  45.  Such  a  sur- 
face as  this  one  (which 
is  convex  upward  along 
the  x-axis  and  concave 

upward  along  the  ?/-axis)  is  commonly  called  a  saddle-back 
and  represents  roughly  the  shape  of  the  surface  of  the 
earth  in  a  mountain  pass. 

The  irregular  surface  of  the  earth  is  sometimes  repre- 
sented on  maps  by  horizontal  section  lines  (contour  lines) 
which  make  it  easy  to  find  the  location,  height,  slope, 
etc.,  of  any  hill,  valley,  or  other  surface  character  by 
inspection  of  the  map. 

137.  Use  of  Contour  Maps. — The  lines  of  horizontal 
section  usually  correspond  to  heights  taken  at  equidistant 
intervals,  for  example,  at  20,  40,  60,  80,  .  .  .  feet  above 
mean  sea  level,  and  are  called  contour  lines.  Students 
usually  find  it  most  convenient  to  think  of  them  as  repre- 
senting the  new  shore  lines  that  would  be  formed  if  the 


65 


FIG.  46.  CONTOUR  LINES  OF  A 
SURFACE. — Horizontal  sections  at 
different  levels  of  the  curved  surface 
of  Fig.  45. 


XII 


COORDINATES  IN   THREE  DIMENSIONS 


165 


sea  level  were  to  rise  20  ft.,  40  ft.,  etc.,  above  its  original 
level.  The  uniform  interval,  twenty  feet  in  this  example, 
is  called  the  contour  interval,  and  the  level  of  reference, 


FIG.  47.  CONTOUR  LINES  OF  A  HILL. — The  upper  figure  is  a 
side  view  of  a  hill  80  feet  high,  showing  in  perspective  the  outlines 
that  would  be  produced  if  it  could  be  cut  into  twenty-foot  slices  or 
the  new  shore  lines  that  would  be  produced  if  the  surrounding 
country  could  be  flooded  to  depths  of  20,  40,  60,  and  80  feet. 

The  lower  figure  is  a  set  of  contour  lines  that  represent  a  top  view 
of  the  levels  shown  above.  The  usefulness  of  a  map  is  greatly 
increased  by  having  contour  lines  drawn  or  printed  on  it  in  some 
distinctive  color.  Such  maps  usually  have  the  contours  in  brown, 
rivers  and  lakes  in  blue,  roads,  buildings,  boundaries,  etc.,  in  black. 

The  line  20  is  the  locus  of  all  points  where  the  surface  is  20  ft. 
above  the  plane  of  reference.  Notice  that  the  ground  is  always 
higher  on  one  side  of  a  contour  line  and  lower  on  the  other. 


166  THEORY  OF  MEASUREMENTS 

usually  mean  sea  level,  is  called  the  datum  plane.  The 
diagrams  (Fig.  47)  show  a  vertical  section  of  a  hill  80 
feet  high  and  a  horizontal  plan  of  its  contour  lines. 

Examine  the  contour  map  and  notice  whether  the 
datum  plane  and  the  contour  interval  are  stated  in  the 
margin. 

Find  a  hill  or  other  elevated  area  on  the  map,  and 
notice  the  arrangement  of  the  contour  lines.  What  is 
the  difference,  as  indicated  on  the  map,  between  a  high 
hill  and  a  low  hill?  What  is  the  difference  between  a 
steep  hill-side  and  a  more  gradual  slope? 

The  model  shows  the  100-foot  contour  lines  of  a  hill 
that  is  given  on  the  map.  See  if  you  can  identify  it 
from  the  shape  of  the  horizontal  sections. 

Find  a  brook  that  runs  down  a  hill.  In  what  general 
direction  do  the  contour  lines  cross  the  brook?  Why? 
What  is  the  general  shape  of  the  contour  lines  where 
there  is  a  water-course?  Where  a  hill  sends  out  a  pro- 
jecting buttress  or  ridge  what  is  the  general  contour 
form?  How  is  a  plateau  formation  indicated  by  contour 
lines? 

Why  is  it  that  contour  lines  where  they  run  across  a 
road- way  are  never  as  near  together  as  they  often  are  in 
other  localities?  Find  a  road  (on  the  map)  that  appears 
to  have  been  purposely  so  constructed  as  to  cut  the  con- 
tour lines  at  considerable  intervals  of  distance. 

138.  Construction  of  a  Contour  Map. — Copy  the  fol- 
lowing table  of  altitudes  upon  the  squared  paper  of  your 
notebook,  writing  each  number  with  ink  in  very  small 
figures  directly  on  the  intersection  of  two  of  the  ruled 
lines.  Make  all  the  spaces  between  columns  of  numbers 
equally  wide  (say  3  or  4  squares),  and  space  the  hori- 
zontal lines  of  numbers  at  the  same  distances  as  the 
intervals  between  the  columns.  Omit  one  or  two  lines 


XII 


COORDINATES  IN   THREE  DIMENSIONS 


167 


at  the  bottom  of  the  table  or  one  or  two  columns  at  the 
right-hand  side  rather  than  crowd  the  numbers  close 
together,  if  the  page  of  your  notebook  is  not  large. 


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ALTITUDES  IN  CENTRAL  PHILADELPHIA. — The  data  are  expressed 
in  feet  above  mean  sea  level.  No  datum  is  given  here  for  the  corner 
of  Sixth  and  Market  streets. 


168  THEORY  OF  MEASUREMENTS  §139 

Copy  also  the  dotted  line  that  represents  a  34-ft.  contour, 
noticing  that  it  passes  exactly  through  the  points  marked 
34,  runs  half-way  between  the  points  33  and  35,  runs 
between  32  and  35  twice  as  far  from  the  former  as  from 
the  latter,  etc. 

Let  your  contour  interval  be  3  feet,  and  start  a  31- 
foot  contour  line  somewhere  near  the  center  of  the 
diagram.  Extend  it  carefully  in  both  directions,  remem- 
bering that  all  the  altitudes  close  to  one  side  of  it  must 
be  greater  than  31  and  all  those  near  the  other  side  must 
be  less  than  31.  In  case  it  is  difficult  or  impossible  to 
extend  the  line  further  than  a  certain  point  leave  it  and 
begin  the  construction  of  another  contour  line.  After 
the  easier  lines  have  been  finished  they  will  be  found  of 
considerable  aid  in  helping  to  determine  the  course  of 
the  more  difficult  ones.  Remember  that  a  single  level 
may  be  represented  by  two  or  more  lines  that  do  not 
join;  thus,  there  will  be  a  34-foot  ring  around  the 
37-foot  peak  at  Fifth  and  Race  Streets,  and  this  cannot 
connect  with  the  line  shown  on  the  diagram  because  of 
the  low-lying  ground  between.  If  the  sea  level  were 
raised  34  feet  the  line  indicated  here  would  be  the  new 
shore  line  and  the  34-foot  ring  would  be  the  shore  of  a 
separate  island. 

Draw  new  contour  lines  at  intervals  of  every  three 
feet  until  the  whole  area  is  covered. 

Estimate  the  elevation  at  the  corner  of  Sixth  and 
Market  Streets  by  interpolation  along  Market  Street 
(E.  and  W.),  also  by  interpolation  along  Sixth  Street 
(N.  and  S.)  and  along  two  diagonal  lines  (NE.  and  SW., 
and  NW.  and  SE.).  Finally  decide  for  yourself  the 
most  reasonable  value  for  this  elevation. 

139.  Questions  and  Exercises. — 1.  If  x/a  +  y/b  +  z/c 
=  1  is  known  to  represent  a  plane  surface  what  can 


XII         COORDINATES  IN   THREE  DIMENSIONS          169 

you  prove  about   the  position  of   this   plane   (compare 
§  116)? 

2.  Write  the  equation  of  the  cone  (curved  surface) 
produced  by  rotating  the  straight  line  z  =  3z  around  the 
2-axis  (suggestion:  rotation  changes  each  point  that  was 
located  at  a  distance  of  x  to  the  right  of  the  2-axis  into  a 
horizontal  circle  whose  radius  is  equal  to  that  £-value). 

3.  What  kind  of  a  locus  corresponds  in  general  to 
three  simultaneous  equations? 

4.  Can  two  contour  lines  representing  different  levels 
ever  intersect  each  other  on  a  map?     What  would  be 
true  of  the  earth's  surface  at  the  point  of  intersection? 

5.  Show  that  the  curves  RCAQ  and  GFDP  made  by 
cutting  the  hyperbolic  paraboloid  (Fig.  45)  at  any  dis- 
tance to  the  left  or  the  right  of  the  origin  by  the  plane 
y  =  a  are   " arch- shaped"    parabolas.     Show  that   the 
curves  RHKG  and  QLNP  made  by  limiting  the  surface 
at  the  front  or  back  by  the  plane  x  =  a  are  "  festoon- 
shaped  "  parabolas. 


XIII.     ACCURACY 

Apparatus. — Rectangular  wooden  block  measuring 
about  4X8X8  cm.;  centimetre  and  millimetre  scale; 
one  scale  (of  centimetres  and  millimetres)  and  one  steel 
tape-measure  to  be  used  by  the  whole  class. 

140.  Significant  Figures. — Significant  figures  have  al- 
ready been  defined  as  all  of  those  that  compose  a  number 
except  one  or  more  ciphers  at  the  extreme  left  or  right 
which  may  be  necessary  to  express  the  order  of  magni- 
tude of  the  number  (i.  e.,  to  locate  the  proper  position 
of  the  decimal  point)  but  are  not  needed  in  any  other  way 
for  indicating  its  value.  For  example,  .003803,  3.803, 
and  3803000  have  four  significant  figures  each;  provided 
that  the  last  one  is  a  " round  number"  (i.  e.,  is  meant  to 
be  accurate  only  to  a  whole  number  of  thousands). 
The  ambiguity  in  the  last  case  can  best  be  avoided  by 
writing  3.803  X  106;  and  the  same  three  numerical 
values  could  all  be  written  with  seven-figure  accuracy 
by  expressing  them  as  3.803000  X  1C)-3,  3.803000,  and 
3.803000  X  106  respectively. 

It  is  usually  understood,  in  any  kind  of  careful  scien- 
tific work,  that  a  number  is  never  written  with  too  many 
significant  figures,  which  would  appear  to  give  it  an 
unwarranted  degree  of  accuracy;  nor  with  too  few  signifi- 
cant figures,  which  would  mean  a  neglect  of  the  full 
extent  of  the  accuracy  that  had  been  obtained.  In 
other  words,  figures  are  known  to  be  correct  as  far  as 
they  are  stated  and  are  unknown  for  all  notational 
"places"  beyond.  For  example,  if  1  yard  is  found  to 
be  83.8  cm.,  the  best  value  of  1  foot  that  can  be  calcu- 
lated from  this  determination  is  1  ft.  =  27.9  cm.  and 

170 


XIII  ACCURACY  171 

the  ordinary  arithmetical  result,  1  ft.  =  27.93333  cm., 
is  not  only  unjustified,  but  in  this  case  is  positively  wrong 
although  the  statement  that  1  yard  =  83.8  inches  is  right. 
Which  four  of  the  following  values  for  e  are  correct 
and  which  six  are  incorrect:  2.71828182;  2.71828183; 
2.718281;  2.7182;  2.7183;  2.7180;  2.7;  2.70;  2.71;  2.72? 

141.  Infinite  Accuracy. — A  number  like  the  ratio  of 
the  diagonal  of  a  square  to  its  side  or  the  ratio  of  circum- 
ference to  diameter  for  any  circle,  which  depends  only 
upon  theoretical  considerations,  can  be  calculated  with 
any  desired  degree  of  accuracy,  but  no  number  whose 
value   depends   upon   the   measurement    of   a   material 
thing  can  be  stated  with  more  than  a  definite  degree  of 
accuracy,  on  account  of  the  limitations  of  the  method  of 
measurement.     If   an   imaginary   circle   is   made   large 
enough  to  enclose  the   Milky  Way  its   circumference, 
measured   in   terms   of   its   radius,    must   be   equal   to 
2  X  3.1415926535897932384626433832795028841971694, 
but  if  it  were  possible  to  measure  a  real  line  of  corre- 
sponding dimensions  the  best  microscope  in  existence 
would  not  enable  us  to  decide  whether  the  value  of  TT 
should  be  as  small  as  3.141592653589793238462643383279 
or  as  large  as  3.141592653589793238462643383280.*    The 
numerical  value  of  TT  has  been  calculated  as  far  as  707  deci- 
mal places,  but  this  of  course  is  not  even  an  approach 
to  perfect  (i.  e.,  infinite)  accuracy. 

142.  Relative  Errors. — If  the  thickness  of  a  lead  pencil 
is  measured  by  holding  it  in  fronq.  of  a  scale  the  separate 

*  Such  a  circle  might  have  a  circumference  of  100,000  "light- 
years,"  or,  say,  1023  centimetres,  considering  the  velocity  of  light 
to  be  30,000,000,000  cm.  per  second  and  the  number  of  seconds  in 
a  year  to  be  over  30,000,000.  If  a  "homogeneous  immersion" 
objective  can  separate  two  points  at  a  distance  of  1  or  2  X  10~5  cm. 
it  could  measure  with  an  accuracy  of  nearly  10~28,  corresponding 
to  28  significant  figures. 


172  THEORY  OF  MEASUREMENTS  §142 

measurements  made  by  a  class  of  students  will  be  likely 
to  vary  as  much  as  0.03  cm.  This  is  nearly  four  per 
cent  of  the  distance  measured.  If  the  length  of  a  six- 
foot  table  top  is  determined  with  an  ordinary  steel 
tape  measure  the  results  that  are  obtained  will  be  apt 
to  vary  two  or  three  millimetres.  This  is  about  J^  per 
cent,  of  the  distance  measured.  In  the  latter  case  the 
error  is  about  ten  times  as  great,  considered  as  an 
isolated  length,  as  it  is  in  the  former.  Considered  in 
relation  to  the  thing  measured,  however,  0.16  per  cent 
is  much  smaller  than  4  per  cent.  Since  the  measurement 
of  the  table  is  obviously  a  more  accurate  process  than 
that  of  the  lead  pencil  it  will  be  evident  that  accuracy 
is  a  matter  of  relative  size,  not  of  absolute  size.  To 
state  that  a  measurement  is  uncertain  by  4  per  cent 
means  something  definite,  but  the  statement  that  a 
measurement  is  correct  "  within  0.03  cm. "  tells  us  nothing 
about  its  accuracy,  if  nothing  is  stated  about  the  length 
measured. 

Measure  a  lead  pencil  and  a  table  in  the  manner 
described.  The  other  members  of  the  class  are  to  meas- 
ure the  same  objects  with  the  same  ruler  and  the  same 
tape.  Report  the  measurements  to  the  instructor  to 
be  tabulated.  Then  determine  their  maximum  dis- 
crepancy, both  in  centimetres  and  as  two  percentages. 

By  using  the  most  refined  methods  it  is  possible  to 
measure  a  distance  of  several  miles  with  remarkable 
accuracy.  If  an  accurate  steel  tape  is  used,  which  is 
stretched  by  a  measured  force  when  it  is  at  a  carefully 
determined  temperature,  it  is  possible  in  the  course  of 
a  few  weeks  to  measure  a  base  line  for  surveying  purposes 
with  an  error  of  about  one  unit  out  of  500000.  Make 
a  rough  mental  calculation  (assume  5000  ft.  =  1  mile) 
of  what  such  an  error  would  amount  to  in  measuring  a 


XIII  ACCURACY  173 

distance  of  ten  miles.     Is  it  larger  or  smaller  than  the 
0.03  cm.  of  the  lead-pencil  measurement? 

Does  the  accuracy  with  which  a  number  is  stated 
depend  upon  the  number  of  decimal  places  to  which  it  is 
carried  out,  or  upon  the  number  of  significant  figures 
which  it  contains? 

143.  Uncertain  Figures. — The  figure  that  follows  the 
last  trustworthy  figure  of  a  measurement  may  be  un- 
certain to  the  extent  of  two  or  three  units,  and  yet  its 
approximate  value  may  be  definite  enough  to  make  one 
hesitate  to  discard  it.     This  would  probably  be  the  case 
if  the  student  attempted  to  estimate  hundredth*  when 
using  a  scale  of  whole  centimetres  without  millimetre 
graduations.     In  such  cases  it  is  better  to  retain  the 
doubtful  figure,  keeping  in  mind  however  the  fact  that 
the  result  obtained  from  it  in  any  calculation  will  be 
liable  to  have  one  uncertain  figure  also. 

Another  case  in  which  it  is  sometimes  desirable,  to 
keep  a  single  uncertain  figure  is  when  a  higher  degree  of 
accuracy  is  made  possible  by  retaining  it  during  the 
process  of  a  calculation,  although  it  is  eventually  dis- 
carded in  stating  the  final  result.  This  is  illustrated  in 
the  treatment  of  the  " figure  last  canceled"  in  the 
'processes  of  abridged  multiplication  and  division. 

144.  Superfluous  Accuracy. — An  appreciation  of  the 
degree  of  accuracy  that  is  required  in  particular  cases 
often  makes  it   possible   to   avoid   needless   trouble  in 
measuring  or  calculating.     If  the  dimensions  of  a  rec- 
tangular block  cannot  be  measured  with  more  than  three- 
figure  accuracy  its  density  can  be  obtained  by  weighing 
it  merely  with  three-figure  accuracy,  and  the  determina- 
tion of  the  fourth  figure  of  its  mass  or  weight  will  not 
enable  a  better  calculation  of  its  density  to  be  made. 

Obtain  the  dimensions  of  a  rectangular  wooden  block 


174  THEORY  OF  MEASUREMENTS  §145 

as  accurately  as  possible  with  a  scale  of  centimetres  and 
millimetres,  estimating  tenths  of  a  millimetre.  Calculate 
its  volume,  using  the  proper  number  of  significant  figures, 
and  then  find  its  density  by  weighing  it  and  dividing 
mass  by  volume. 

145.  Finer  Degrees  of  Accuracy. — A  measurement 
may  happen  to  be  known  with  an  accuracy  greater  than 
can  be  expressed  by  three  significant  figures  and  yet 
not  with  four-figure  accuracy;  that  is,  the  step  from 
any  degree  of  accuracy  to  a  ten-fold  greater  degree  may 
be  too  great  to  be  suitable  in  all  cases.  For  example, 
a  period  of  time  equal  to  125f  seconds  may  have  been 
measured  more  accurately  than  to  the  nearest  whole 
number  of  seconds  and  yet  may  not  have  been  deter- 
mined closely  enough  to  enable  tenths  of  a  second  to  be 
stated.  In  such  cases  common  fractions  may  be  used 
instead  of  decimals;  and  a  significant  zero  can  be  em- 
ployed in  such  a  form  as  "125f  seconds"  without  any 
lack  of  intelligibility,  meaning  of  course  "  between  124.9 
and  125.1  seconds/'  or  "  nearer  to  125  seconds  than  to 
124£  or  to  125£."  It  is  usually  more  satisfactory,  how- 
ever, to  state  a  numerical  value  for  a  measurement  and 
then  affix  a  statement  of  its  uncertainty  in  the  form  of  a 
percentage  or  otherwise. 

What  would  you  understand  to  be  the  largest  possible 
error  in  a  measurement  stated  to  be  125  seconds?  Ans.: 
0.5  sec.,  or  0.4  per  cent. 

What  would  you  understand  to  be  the  largest  possible 
error  in  a  measurement  stated  to  be  125.0  seconds? 
What  would  it  be  for-  125f  sec.?  Express  the  answers 
both  in  seconds  and  in  percentages. 

Consider  carefully  the  stated  measurement  "22J  cubic 
centimetres."  If  it  had  to  be  written  as  a  decimal  for 
the  sake  of  averaging  it  with  a  set  of  other  measurements 


XIII  ACCURACY  175 

would  you  prefer  to  write  it  22.3  or  22.33,  or  would  you 
round  it  off  to  22?  Explain  why. 

146.  Possible  Error  of  a  Measurement. — Instead  of 
stating  that  a  measurement  may  have  an  error  in  a 
certain  decimal  place  or  significant  figure  although  the 
figures  that  precede  it  are  correct  it  is  usually  better  to 
state  the  possible  error  in  the  form  of  a  ratio  or  a  per- 
centage. Since  the  figures  of  a  numerical  statement  are 
intended  to  be  significant  and  correct  as  far  as  they  go 
it  is  evident  that  the  statement  cannot  have  an  error 
larger  than  half  of  a  single  unit  in  the  last  decimal  place, 
or  five  units  in  the  place  that  would  follow  the  last  one 
that  is  written.  Accordingly,  a  statement,  like  that  of 
§  56,  that  a  length  which  is  written  174.2  certainly 
cannot  have  an  error  of  more  than  1  out  of  1742,  is  on 
the  safe  side  but  could  be  improved  by  making  it  read 
"not  more  than  \  out  of  1742,  or  1  out  of  3484,  or 
.0003,  or  .03  per  cent." 

Turn  to  your  experimental  determinations  of  sines 
(§  40),  find  the  greatest  possible  error  of  each  as  half  of  a 
single  unit  in  the  last  written  decimal  place,  and  reduce 
this  possible  error  either  to  a  decimal  fraction  (e.  g.,  1 
out  of  25  is  the  same  as  .04)  or  to  a  percentage  (1  out  of 
25  =  4  per  cent),  according  to  which  form  appeals  to  you 
as  being  the  more  expressive.  Then  mark  each  error 
"small,"  "large,"  etc.,  as  given  in  the  table  of  errors 
classified  according  to  size  (see  appendix). 

Notice  that  a  gradation  which  is  as  coarse  as  that 
expressed  by  enumerating  significant  figures  is  unsatis- 
factory in  certain  cases.  The  number  .9624  has  no 
more  significant  figures  than  1.093  but  is  expressed  with 
about  ten  times  as  great  accuracy  as  the  latter.  For 
purposes  of  using  it  in  a  calculation  along  with  1.093  it 
should  be  rounded  off  to  .962.  According  to  the  rules 


176  THEORY  OF  MEASUREMENTS  §147 

given  in  the  chapter  on  small  magnitudes  1.093  X  .9624 
would  equal  1.  +  .093  -  (1  -  .9624)  =  1.  +  .093  -  .0376 
=  1.0554.  Explain  why  this  result  is  unjustifiable,  and 
show  the  fallacy  in  the  process  of  obtaining  it. 

147.  Possible  Error  After  a  Calculation. — The  possible 
error  of  the  sum  of  two  or  more  measurements  is  equal 
to  the  sum  of  their  individual  possible  errors. 

A  table  is  found  by  measurement  to  be  44.3  cm.  higher 
than  a  bench  which  has  a  height  of  42.5  cm.  from  the 
floor.  If  the  third  significant  figure  of  each  measurement 
may  have  a  possible  error  of  half  a  unit  what  limits  can 
be  assigned  for  the  height  of  the  table  from  the  floor. 
Ans.:  between  86.7  cm.  and  86.9  cm. 

The  possible  error  of  the  difference  between  two  measure- 
ments is  equal  to  the  sum  (not  the  difference)  of  their 
possible  errors. 

A  table  is  51.1  cm.  lower  than  a  shelf  which  is  between 
137.9  cm.  and  140.1  cm.  above  the  floor.  How  high 
must  the  table  be?  Ans. :  from  88.75  cm.  to  89.05  cm. 

Would  it  be  correct  to  answer  the  last  example,  "from 
88.8  cm.  to  89.0  cm."? 

The  possible  error  of  the  product  of  the  two  measure- 
ments mi  ±  e\  and  m-,  ±  e2  will  be  found  by  ordinary 
multiplication  to  be  m\e<i  +  m^e\  under  ordinary  circum- 
stances in  which  the  error  is  small  (§  70)  compared  with 
the  measurements  themselves,  so  that  0162  is  negligible 
(§  72).  It  will  be  instructive,  however,  to  re-write  the 
measurement  and  its  possible  error,  m  ±  e,  in  the  form 
m(l  d=  e/m)  before  performing  the  multiplication.  The 
fraction  e/m  will  now  represent  the  relative  error  (§  54). 
The  product  will  be  m\(\  d=  ei/Wi)  X  w2(l  ±  e2/w2) 
=  raira2(l  d=  ei/mi  d=  e2/m2);  *•  e-»  the  possible  percentage 
error  of  a  product  is  equal  to  the  sum  of  the  possible  per- 
centage errors  of  its  factors. 


XIII  ACCURACY  177 

A  rectangular  surface  measures  50  cm.  X  20  cm.  Cal- 
culate (50  ±  0.5)  X  (20  ±  0.5),  and  show  that  the  pos- 
sible error  of  the  area  is  35  cm2;  then  find  the  relative 
Value  of  0.5  to  50  and  that  of  0.5  to  20,  and  see  whether 
their  sum  is  the  same  as  the  ratio  of  35  cm2  to  50  cm. 
X  20  cm. 

Similarly,  the  possible  percentage  error  of  a  quotient  is 
equal  to  the  sum  of  the  possible  percentage  errors  of  the 
divisor  and  dividend;  for  Wi(l  ±  ei/Wi)  -r-  w2(l  ±  e^/niz) 
is  equal  to  (1  ±  eijm\  d=  62/^2)^1/^2. 

These  rules  are  equally  applicable  in  cases  where  one 
of  the  two  quantities  employed  has  no  error  whatever. 
If  a  50-foot  tape  measure  has  a  possible  error  of  a  tenth 
of  a  foot  (0.2  per  cent)  then  there  may  be  an  error  of 
three  tenths  of  a  foot  in  using  it  to  measure  a  distance 
of  150  feet.  If  the  measured  diameter  of  a  circle  is 
uncertain  by  3  per  cent  then  the  calculated  circum- 
ference will  be  liable  to  an  error  of  3  per  cent. 

148.  "Probable"  Error.— The  magnitude  of  a  " pos- 
sible error"  is  not  actually  as  definite  as  it  is  usually 
assumed  to  be.  There  may  be  a  high  degree  of  prob- 
ability that  a  measurement  has  no  error  greater  than  half 
a  unit  in  the  last  decimal  place  that  is  written,  but  this 
can  never  be  so  high  as  to  amount  to  an  absolute 
certainty. 

The  most  that  can  be  said  is  that  a  certain  large 
range  for  the  error  of  a  measurement  is  very  improb- 
able, narrower  limits  of  error  are  less  improbable,  that 
an  error  is  within  a  still  smaller  range  may  be  probable 
rather  than  improbable,  that  the  error  has  at  least  a 
certain  minute  magnitude  may  be  so  highly  probable 
as  to  be  almost  a  certainty,  etc.  Without  taking  up  a 
quantitative  discussion  of  the  relative  likelihood  that 
the  magnitude  of  a  particular  error  will  fall  within  given 
13 


178  THEORY  OF  MEASUREMENTS  §150 

limits  we  need  notice  here  only  that  the  most  convenient 
degree  of  probability  that  we  can  specify  is  that  of  a 
half-way  point  between  probability  and  improbability. 
For  a  certain  limited  class  of  error  it  is  possible  to  state 
that  a  particular  error  is  "just  as  likely  as  not"  to  be 
within  certain  numerical  boundaries,  i.  e.,  that  the 
error's  chance  of  being  smaller  than  the  stated  limit  is 
just  equal  to  its  chance  of  being  larger  than  the  limiting 
value.  Boundaries  of  this  sort  are  sometimes  called 
probable  errors,  and  will  be  considered  in  a  later  chapter. 
At  present  all  that  needs  to  be  noticed  in  regard  to  them 
is  that  they  are  not  " probable"  in  the  sense  of  being 
highly  probable  values  nor  are  they  even  " errors"  except 
in  a  restricted  sense  of  the  word,  so  that  the  accepted 
name  is  apt  to  prove  a  misleading  one. 

149.  Accuracy  Required  in  Special  Cases. — When  only 
a  certain  degree  of  accuracy  is  needed  there  is  nothing 
to  be  gained  by  attempting  a  greater  degree,  and  there 
is  usually  a  considerable  loss  of  time  and  labor  if  calcula- 
tions are  carried  out  to  more  significant  figures  than  can 
be  utilized.     For  a  great  many  scientific,  engineering, 
and    commercial    calculations    three-figure    accuracy   is 
sufficient,    and  in   such    cases    the   slide    rule    can    be 
used  more  readily  than  a  table  of  logarithms.     Four- 
figure   accuracy  is   attained   by  the   use  of   four-place 
logarithm  tables,  and  is  sufficient  for  most  of  the  calcula- 
tions   of    physics    and    chemistry,    although    five-figure 
accuracy  is  needed  for  certain  measurements,  e.  g.,  deter- 
minations of  mass,  periods  of  time,  barometric  pressures, 
etc.     Six-figure  accuracy  is  sufficient  for  all  surveying 
except  the  most  precise  geodetic  measurements.     Seven- 
figure  accuracy  is  also  needed  for  certain  astronomical 
calculations. 

150.  Questions  and  Exercises. — 1.  An  expression  like 


XIII  ACCURACY  179 

"  four-and-a-half-figure  accuracy"  is  sometimes  seen, 
Write  down  your  opinion  of  what  this  phrase  means, 
expressed  as  a  percentage;  and  state  your  reasons  for 
choosing  this  number. 

2.  Explain  why  the  possible  error  of  a  difference  is 
equal  to  the  sum  (instead  of  difference)  of  the  errors  of 
its  components. 

3.  If  a  wooden  block  measures  250  cm3,  with  an  un- 
certainty of  5  cm3,  and    weighs  2.00  X  102  gm.,   how 
many    gm./cm3    does    the    uncertainty    of    its    density 
amount  to?     Ans.  :  sixteen  or  eighteen  hundredths  of  a 
unit.     What  can  you  say  of  the  effect  of  the  uncertainty 
of  the  weight? 

4.  Each  volume  of  a  ten-volume  encyclopedia  has  2.0 
inches  thickness  of  leaves  between  two  covers  that  are 
each  0.2  inch  thick.     If  the  number  of  pages  per  volume 
is  liable  to  vary  by  2  per  cent  and  the  thickness  of  the 
cover  by  1  per  cent,  what  is  the  maximum  length  of 
shelf-space   that   may   be   needed   for   the   set?     Ans.: 
24.44  inches. 

If  a  calculated  measurement  M  is  equal  to  lOwi  +  20m2 
what  will  its  possible  error  E  amount  to  in  terms  of  the 
possible  errors  e\  and  e2  of  mi  and  m2  respectively? 
Suggestion  : 


+  where 


5.  The  quotient  in  §  147  should,  strictly  speaking,  be 
(1  =t  ei/mi  =F  e2/w2)wi/ra2.     Explain  why  the  inversion  of 
the  double  sign  is  unnecessary  in  this  connection. 

6.  Write  in  your  notebook  a  brief  resume  of  all  of  the 
important  points  in  chapters  iv  and  xiii. 


XIV.     THE   PRINCIPLE   OF    COINCIDENCE 

Apparatus. — Two    metre    sticks;     cardboard;     knife; 
rectangular  wooden  block;  slide  rule. 

151.  Effect  of   Magnitude   upon  Accuracy. — With   a 
scale  of  millimetres  measure  a  length  of  one  inch  as 
accurately   as   possible,    writing   down   the   number   of 
whole   centimetres,    additional   whole   millimetres,    and 
estimated   additional  tenths   of  a  millimetre   which   it 
contains.     Measure  a  length  of  two  inches  in  the  same 
way  and  divide  the  result  by  two.     Calculate  one  tenth 
of  the  carefully  measured  length  of  an  interval  of  ten 
inches  and  compare  with  the  two  other  results.     Notice 
that  a  large  quantity  can  usually  be  measured  more 
accurately  than  a  small  one.     If  the  method  of  measure- 
ment gives  the  same  absolute  accuracy  in  two  different 
cases    (for  example,   no   error   greater   than   .005   cm.) 
the  ratio  of  error  to  measurement  will  naturally  be  a 
smaller  relative  error  just  in  proportion  as  the  measure- 
ment itself  is  greater.     It  is  for  this  reason  that  the  metre 
was  chosen  for  a  standard  of  length  (§  19)  instead  of  the 
centimetre,  and  the  kilogram  for  a  standard  of  mass 
instead  of  the  unit,  a  gram. 

152.  Measurement  by  Estimation. — In  order  to  make 
a  more  accurate  determination  of  the  length  of  an  inch 
lay  one  metre  stick  on  the  table  with  the  metric  gradu- 
ations upward  and  place  the  other  beside  it  with  the 
graduation  in  inches   and  tenths  of  an  inch  upward. 
See  that  the  scales  are  in  close  contact  and  note  down 
the  number  of  centimetres  and  hundredths  that  happens 
to  be  opposite  the  mark  1  inch  on  the  other  scale.     Note 
also  the  indication  of  the  mark  31  inches.     Move  one 

180 


XIV 


THE  PRINCIPLE  OF  COINCIDENCE 


181 


scale  along  the  other  at  random  and  repeat  the  obser- 
tions,  unless  otherwise  directed,  until  ten  sets  have  been 
taken.  The  distance  measured  should  always  be  the 
same,  30  inches,  but  it  should  be  measured  at  other 
places,  such  as  the  mark  4  and  the  mark  34.  Never  use 
the  mark  0  at  the  very  end  of  the  stick,  as  it  is  often 
inaccurate  on  account  of  wear. 
Tabulate  the  results  in  a  form  like  the  following : 

MEASUREMENT  OF  AN  INCH  BY  THE  METHOD  OF  ESTIMATION. 


1st  line 

2d  line 

Interval 

Cm.  scale  .... 
Inch  scale  .... 

20.05 
1.00 

45.50 
11.00 

25.45 

10.00 

Cm 

50.00 
2.70 

75.47 
12.70 

25.47 

10.00 

Inch  ........ 

Cm 

10.06 
1.50 

35.46 
11.50 

25.40 

10.00 

Inch  

• 

• 

Sum  

254.02 
25.402 

100.00 
10.000 

Average  

Length  of  an  inch : 


cm. 


153.  Measurement  by  Coincidence. — In  the  method 
of  measurement  by  estimation  one  or  more  equal  intervals 
are  measured  in  the  customary  manner,  tenths  of  the 
smallest  scale-divisions  being  determined  by  a  mental 
estimate.  For  the  method  of  measurement  by  coinci- 
dence a  series  of  equal  quantities  is  necessary.  If  these 
can  be  directly  compared  with  a  series  of  equal  scale- 
units  it  will  usually  be  possible  to  observe  that  some  whole 


182  THEORY  OF  MEASUREMENTS  §153 

number  of  the  equal  quantities  is  of  precisely  the  same 
magnitude  as  some  other  whole  number  of  scale-divisions. 
For  example,  the  diagram  (Fig.  48)  shows  in  two  different 

l  .     1|0 .  210  i  C  8|0 i 

I  M  i,  i.l  ,i  i  i  i  I  i  i  .1  .1     i  i  i  i  I  i  i.  i.  i.l  .1  ,i  i.  i,  I 


T-TT 


i ',',', '.'  i1  u '.','.'  i  I'.y.'M  '.','.'.'  i 

210  3  0 


I,  1,1 


i  'i'  i    M   'I"T 


Ho 

FIG.  48.  THE  METHOD  OF  COINCIDENCES. — Each  of  the  un- 
known intervals  is  seen  to  be  a  trifle  less  than  1  cm.,  but  the  fact 
that  there  is  no  distinguishable  difference  between  13  intervals  and 
11  cm.  shows  that  the  length  of  the  unknown  interval  must  be 
11/13  of  a  centimetre. 

locations  (c)  that  13  of  the  unknown  intervals  are  equal 
to  11  of  the  centimetre  units.  Accordingly,  the  unknown 
interval  must  be  11/13  cm. 

Set  the  two  scales  so  that  any  whole  number  of  inches 
near  one  end  of  one  metre  stick  is  exactly  opposite  some 
whole  number  of  centimetres  on  the  other.  Hold  the 
two  sticks  firmly  together  and  look  again  to  see  that  the 
coincidence  is  exact.  Do  not  be  satisfied  unless  you  are 
unable  to  say  whether  the  upper  mark  is  a  little  toward 
the  right  or  toward  the  left  of  the  one  below  it.  Without 
allowing  the  two  scales  to  slip  find  another  place,  at 
least  30  cm.  distant  from  this  point  and  preferably 
further,  where  there  is  another  exact  coincidence,  this 
time  between  any  centimetre  or  millimetre  graduation 
and  any  graduation  of  inches  or  tenths.  Try  to  decide 
whether  the  coincidence  is  exact  or  whether  the  imaginary 
central  axis  of  one  line  lies  a  little  beyond  the  other, 
and  if  the  coincidence  is  faulty  choose  a  better  one 
elsewhere. 

Record  the  results  as  in  the  specimen  table,  re-set 
the  two  scales,  and  repeat  until  10  determinations  have 
been  made.  Shift  the  position  of  the  first  coincidence, 
as  was  done  in  the  method  of  estimation;  also,  take  care 


XIV 


THE  PRINCIPLE  OF  COINCIDENCE 


183 


that  the  interval  between   coincidences  is   not  of  the 
same  length  each  time. 

MEASUREMENT  OF  AN  INCH  BY  THE  METHOD  OF  COINCIDENCES. 


1st  line 

2d  line 

interval 

cm.  in  1  in. 

Cm..  . 

90.000 

35.400 

54.600 

In  

4.000 

25.500 

21.500 

2.5395 

Cm.. 
In  

15.000 
6.000 

71.400 
28.200 

56.400 
22.200 

2.5405 

• 

• 

• 

• 

Averaae  .  . 

2.5401 

It  is  not  absolutely  accurate  to  assume  that  the  co- 
incidences are  always  exact  to  a  hundredth  of  the  smallest 
graduation  of  the  scale,  but  the  lack  of  perfect  coincidence 
is  usually  perceptible  even  if  the  discrepancy  is  much 
less  than  a  tenth  of  a  scale-division.  It  will  very  rarely 
exceed  a  hundredth  of  a  smallest  subdivision  if  the  work 
is  carefully  done. 

154.  The  Vernier. — Rule  a  straight  line  along  a  strip 
of  cardboard  5  or  10  cm.  wide  and  about  20  cm.  long 
(Fig.  49).  As  shown  in  the  diagram,  lay  off  a  scale  of 


f 

i.     "T 

i    i   i    i    1  i    , 

I 

i  i 

\ 

B    I    i    i    i    '    |    i 

\            lo                      5 

\  <sealt 

'Mill 
»lo 

i  i  i  i 

15 

'T 

FIG.  49.  MODEL  OF  A  VERNIER  CALIPER. — The  diagram  can 
be  drawn  on  cardboard,  cut  out,  and  used  for  fairly  accurate  measure- 
ments. 


184  THEORY  OF  MEASUREMENTS  §154 

centimetres  along  the  lower  side  of  the  line;  along  the 
upper  side  lay  off  a  series  of  ten  intervals,  making  each 
one  0.9  cm.  long,  so  that  their  combined  length  will  be 
just  9  cm.  Rule  a  perpendicular  line,  AB,  a  little  to 
the  left  of  the  common  zero  point,  and  mark  the  words 
"scale"  and  " vernier"  as  indicated.  See  that  your 
work  corresponds  in  every  respect  to  Fig.  49,  except 
that  the  letters  A,  B,  and  C  are  not  marked,  and  the 
scale  of  centimetres  may  be  longer  or  shorter  than  shown 
here.  Observe  that  the  mark  1  on  the  vernier  is  one 
tenth  of  a  centimetre  to  the  left  of  the  mark  1  on  the 
centimetre  scale,  2  on  the  vernier  is  two  tenths  to  the 
left  of  2  on  the  scale,  etc.  Put  the  card  on  a  larger 
piece  of  cardboard,  to  avoid  marring  the  table,  and  cut 
from  C  to  B,  but  no  further,  also  from  A  to  B.  The 
result  will  be  a  cardboard  model  of  a  vernier  caliper. 
Notice  that  the  zero  of  the  vernier  acts  as  a  pointer  that 
indicates  0  cm.  when  the  jaws  (AB)  of  the  caliper  are  no 
distance  apart;  consequently  it  must  give  the  correct 
reading  on  the  centimetre  scale  when  the  jaws  are  slid 
apart  to  any  given  distance. 

Holding  the  body  of  the  vernier  caliper  stationary  on 
the  table  move  the  vernier  slightly  to  the  right  so  that 
its  line  number  1  coincides  with  the  1-centimetre  mark; 
then  move  it  further  until  2  and  2  coincide;  then  3  and 
3;  etc.  When  the  tenth  division  of  the  vernier  has  been 
brought  into  coincidence  with  a  line  on  the  centimetre 
scale  it  will  be  found  that  the  zero  of  the  vernier  stands 
at  the  mark  1  cm.  and  the  jaws  of  the  caliper  are  sepa- 
rated by  a  distance  of  a  single  centimetre.  From  the 
construction  of  the  apparatus  it  will  be  clear  that  when 
any  division  of  the  vernier  (say  number  3)  coincides 
with  any  division  of  the  lower  scale  the  jaws  of  the  caliper 
must  be  separated  by  a  corresponding  number  (3)  of 


XIV  THE  PRINCIPLE  OF  COINCIDENCE  185 

tenths  of  a  centimetre  more  than  some  whole  number  of 
centimetres. 

Separate  the  jaws  of  this  model  vernier  caliper  by  a 
distance  of  eight  and  a  half  centimetres,  as  nearly  as 
you  can  by  estimation;  then  look  at  the  vernier  and 
notice  that  while  its  zero  is  beyond  the  mark  8  of  the 
centimetre  scale  the  division  5  of  the  vernier  coincides 
with  some  division  (no  matter  which  one)  of  the  main 
scale.  The  reading  is  accordingly  8.5  cm.  In  the  same 
way  read  the  indicated  length  when  the  caliper  is  set 
at  random,  and  repeat  the  process  until  you  are  per- 
fectly familiar  with  it. 

155.  Use  of  the  Vernier  Caliper. — Hold  the  two  parts 
of  the  caliper  in  alignment  by  pinching  the  line  BC 
between  the  fore-finger  and  thumb  of  the  left  hand, 
holding  the  lower  part  tightly  but  allowing  the  upper 
part  to  slide;  at  the  same  time  hold  the  same  line 
between  the  fore-finger  and  thumb  of  the  right  hand, 
allowing  them  to  slide  on  the  lower  part  while  grasping 
the  upper  part  firmly.  This  will  allow  the  model  to  be 
accurately  applied  to  any  object  that  is  to  be  measured. 
Use  the  apparatus  to  make  twelve  measurements  of  the 
thickness  of  the  wooden  block  at  different  positions 
around  its  edge.  Write  down  each  measurement  as 
soon  as  it  is  obtained,  and  do  not  be  disconcerted  by 
the  fact  that  many  of  them  may  be  identical.  Find  the 
average  thickness  of  the  block,  carrying  the  result  out  to 
one  more  decimal  place  than  the  individual  measure- 
ments, and  keep  it  for  later  reference. 

If  a  vernier  caliper  is  arranged  to  give  tenths  of  a 
scale-division  it  is  possible  to  make  a  reading  of  half  a 
tenth,  or  a  twentieth,  in  case  two  adjacent  lines  on  the 
vernier  show  equal  and  opposite  deviations  from  co- 
incidence with  two  lines  of  the  scale.  In  this  way  an 


186  THEORY  OF  MEASUREMENTS  §157 

ordinary  barometer  scale  of  twentieths  of  an  inch,  pro- 
vided with  a  25-division  vernier  that  gives  500ths,  can 
be  read  accurately  to  single  thousandths  of  an  inch. 

156.  Slide-Rule  Ratios. — It  has  already  been  seen  that 
setting  a  slide  rule  for  the  ratio  TT  and  picking  out  a  point 
of  exact  coincidence  will  usually  give  a  numerical  value 
for  TT  that  has  more  than  three-figure  accuracy,  i.  e., 
greater  accuracy  than  can  be  obtained  by  the  ordinary 
process  of  " estimation"  with  the  slide  rule.     Numbers 
like  the  26  and  66  on  the  back  of  the  slide  rule  for  the 
ratio   of  inches   and   centimetres   are  so   chosen   as   to 
give  the  correct  value  to  at  least  as  many  significant 
figures  as  can  be  read  with  the  apparatus  used.     Some 
slide  rules  give  the  same  equivalent  as  50  in.  =  127  cm. 
This  has  the  disadvantage  of  not  being  quite  as  easy  to 
set  on  the  A  and  B  scales  as  on  C  and  D,  but  for  use 
with  a  very  finely  graduated  instrument  or  with  one 
on  which  the  scales  are  20  inches  long  instead  of  10  it  has 
the  advantage  of  greater  accuracy,  for  66.0000  -r-  26.0000 
=  25.3816  while  127.0000  -s-  50.00000  =  25.40000.     The 
former  ratio  is  correct  to  three  significant  figures  only, 
while  the  latter  is  correct  to  five  (§  53). 

157.  Questions  and  Exercises. — 1.  Consider  carefully 
what  is  meant  by  the  expression  "the  principle  of  co- 
incidence," and  then  write  a  definition  of  it  in  your  own 
words.     See  that  it  is  a  definition  of  a  principle,  not  of  a 
process  or  of  a  condition. 

2.  What  is  the  significance  of  the  fact  that  the  second 
point  of  "coincidence"  in  Fig.  48  is  not  as  precise  as  the 
first,  nor  the  third  as  good  as  the  second? 

3.  Suppose  the  time  of  vibration  to  be  just  7/8  as 
long  for  one  pendulum  as  for  another.     How  could  you 
verify  the  fact  by  the  method  of  coincidences. 

4.  If  you  take  just  seventeen  steps  while  a  man  who 


XIV  THE  PRINCIPLE  OF  COINCIDENCE  187 

is  walking  beside  you  takes  just  eighteen  how  long  are 
his  steps  in  terms  of  yours,  which  are  considered  of  unit 
length?  How  does  the  frequency  of  his  steps  compare 
with  that  of  yours? 

5.  Explain  why  the  method  of  coincidence  is  more 
exact  than  the  method  of  estimation. 

6.  If  the  numbers  in  the  column  headed  "interval," 
§  153,  are  trustworthy  to  five  significant  figures  how  can 
you  account  for  the  fact  that  even  the  fourth  significant 
figure  fluctuates  in  column  of  centimeters  per  inch? 

7.  If  an  ordinary  vernier  can  be  read  to  half-tenths 
why  can  it  not  be  used  for  other  fractions  of  a  tenth? 

8.  Could  the  upper  scale  in  Fig.  48  be  used  as  a 
vernier  for  the  centimetre  scale?     If  so,  would  it  indicate 
elevenths  of  a  centimetre,   or  thirteenths,   or  neither? 
How  would  its  numbers  be  arranged? 


XV.     MEASUREMENTS  AND   ERRORS 

Apparatus. — A  vernier  caliper;  100  seeds  of  Phaseolus 
(or  other  variates). 

158.  Direct  and  Indirect  Measurements. — An  indirect 
measurement  is  one  that  is  obtained  from  another  meas- 
urement by  means  of  a  calculation.     The  ordinary  proc- 
esses of   measuring   length  or  weight  are  called  direct, 
because  the  unknown  length  is  placed  beside  a  standard 
series  of  multiples  and  fractions  of  the  unit  of  length 
and   is   directly   compared   with   it,    and   an   unknown 
weight  is  directly  balanced  against  a  series  of  known 
weights  until  its  exact  equivalent  is  determined.     An 
example  of  an  indirect  measurement  is  the  usual  method 
of  determining  density.     The  volume  of  an  object  and 
its  mass  are  determined  directly,  or  the  volume  is  cal- 
culated from  other  measurements,   and  its   density  is 
then  obtained  by  making  a  calculation  of  the  ratio  of 
mass  to  volume. 

Are  measurements  of  area  usually  direct  or  indirect? 
Can  they  be  made  in  the  other  way? 

How  could  the  density  of  a  liquid  be  determined  by  a 
direct  measurement? 

How  could  the  density  of  a  solid  be  determined  by  a 
direct  measurement? 

159.  Independent,  Dependent,  and  Conditioned  Meas- 
urements.— Measurements    are    also  classified   as  inde- 
pendent, dependent,  and  conditioned,  and  may  belong  to 
any  one  of  these  classes  whether  they  are  direct  or  indirect. 

Two  or  more  measurements  are  said  to  be  conditioned 
if  there  is  a  theoretical  relationship  between  them, 
which  must  always  hold  good.  The  three  angles  of  a 

188 


XV  MEASUREMENTS  AND  ERRORS  189 

plane  triangle  furnish  an  illustration  of  this  class.  Their 
sum  must  always  be  equal  to  w,  or  180°. 

Measurements  are  said  to  be  dependent  if  one  of  them 
is  allowed  to  influence  or  bias  the  observer  when  making 
a  later  measurement  of  the  same  quantity.  With  accu- 
rate determinations  this  effect  is  so  hard  to  avoid,  even 
if  the  observer  has  the  best  of  intentions,  that  it  is  always 
advisable  to  guard  against  it  by  some  such  device  as 
hiding  the  scale  of  an  apparatus  from  view  until  after 
the  indicating  mark  has  been  set  in  the  position  that  has 
to  be  read.  Measurements  are  also  dependent  if  any 
essential  step  in  making  them  is  not  repeated  in  succes- 
sive determinations  but  is  assumed  to  have  its  effect 
remain  unchanged  during  the  series.  Thus,  in  making 
independent  measurements  by  the  method  of  coincidences 
(§  153)  the  scales  of  length  were  not  held  in  one  position 
while  several  coincidences  were  found,  but  were  reset 
after  each  determination. 

It  is  customary  to  use  the  term  independent  only 
for  measurements  that  are  at  the  same  time  neither 
dependent  nor  conditioned. 

If  the  density  of  an  object  should  be  determined  by  a 
direct  measurement  would  its  mass,  density,  and  volume 
be  independent,  dependent,  or  conditioned? 

160.  Harmony  and  Disagreement  of  Repeated  Meas- 
urements.— If  the  same  object  is  measured  several  times 
in  succession  the  measurements  will  in  general  differ 
from  one  another,  but  the  differences  will  tend  to  be 
small  under  either  of  two  different  circumstances:  (a)  if 
the  quantity  is  of  a  sharply  defined  character,  or  (6)  if 
the  method  of  measurement  is  coarse  or  crude.  Thus, 
if  a  length  of  woolen  cloth  were  compared  with  an  accu- 
rate millimetre  scale  it  would  probably  be  found  difficult 
to  measure  it  the  same  twice  in  succession;  but  if  the 


190  THEORY  OF  MEASUREMENTS  §160 

same  length  of  steel  rail  were  clamped  on  rigid  supports 
and  measured  at  a  constant  temperature  it  might  be 
hard  to  obtain  two  measurements  that  would  differ. 
The  length  of  one  object  would  be  a  poorly  defined 
quantity;  that  of  the  other,  a  sharply  defined  quantity. 

Although  sharpness  of  definition  of  the  quantity  to 
be  investigated  means  that  repeated  measurements  will 
tend  to  harmonize  closely,  it  should  be  carefully  noted 
that  accuracy  of  the  methods  or  means  of  measurement 
has  just  the  opposite  result;  it  is  the  rough  methods  of 
measurement  that  make  repeated  determinations  iden- 
tical with  one  another,  and  the  refined  methods  that 
show  discrepancies.  Two  similar  1-lb.  weights  may 
appear  to  have  precisely  the  same  mass  on  a  rough 
balance  and  yet  differ  when  weighed  on  a  more  carefully 
constructed  one.  If  the  heavier  weight  should  then  be 
filed  down  just  enough  to  make  it  equal  to  the  lighter  one 
a  test  with  a  delicate  chemical  balance  might  show  not 
only  that  the  two  were  still  unequal  but  even  that  one 
of  them  alone  would  not  weigh  the  same  amount  twice 
in  succession. 

The  general  statement  can  be  made,  then,  that  an 
accurately  defined  quantity  or  a  coarse  method  of  meas- 
urement will  result  in  a  series  of  determinations  being 
harmonious  or  identical,  while  a  poorly  defined  quantity 
or  an  accurate  method  of  measurement  will  cause  the 
results  to  disagree  or  diverge. 

Which  measurements  do  you  think  would  be  most  apt 
to  show  variations  among  themselves,  the  measurements 
of  the  wooden  block  which  you  made  with  the  cardboard 
caliper,  or  measurements  of  the  same  block  made  with  a 
steel  vernier  caliper? 

It  is  a  general  truth  that,  no  matter  how  sharply 
defined  a  quantity  may  be,  the  use  of  the  most  precise 


XV  MEASUREMENTS  AND  ERRORS  191 

methods  will  result  in  successive  equally  careful  measure- 
ments of  it  differing  perceptibly  from  one  another,  al- 
though the  differences  may  be  very  small.  Thus,  the 
most  accurate  determinations  of  the  length  of  a  national 
prototype  meter  (§  19)  would  be  apt  to  differ  by  several 
tenths  of  a  micron. 

If  equally  careful  measurements  of  the  same  quantity 
fail  to  agree  and  there  is  no  reason  for  preferring  one 
rather  than  another  it  necessarily  follows  that  the  true 
magnitude  of  a  measured  quantity  is  always  unknown. 
Accordingly  the  various  approximate  measures  are  sum- 
marized, for  actual  use,  by  stating  their  average  (the 
sum  of  n  numbers,  divided  by  n)  or  some  other  repre- 
sentative value,  instead  of  by  choosing  one  of  them  at 
random. 

161.  Errors  of  Measurement. — The  error  of  a  meas- 
urement is  the  amount  by  which  it  differs  from  the  true 
value  of  the  quantity  which  is  measured.  If  the  true 
value  is  always  unknown  the  error  must  likewise  be 
unknown.  Such  errors,  however,  can  be  discussed  the- 
oretically, and  in  this  way  much  can  be  learned  about 
the  best  manner  of  dealing  with  them. 

Errors  are  usually  classified  as  constant  and  accidental, 
but  what  are  known  as  mistakes  really  belong  in  a  sepa- 
rate class  by  themselves.  Constant  errors  affect  all  the 
measurements  of  a  series  in  the  same  manner  or  in  the 
same  direction.  Accidental  errors  are  small  errors  that 
make  one  measurement  a  trifle  too  large  and  another 
too  small,  but  do  not  tend  to  bias  the  average  result. 
Mistakes  are  occasional  errors  that  are  due  to  a  lack  of 
mental  alertness  on  the  part  of  the  observer. 

CLASSIFICATION  OF  ERRORS. 

CONSTANT: 

THEORETICAL:   usually  calculable,  such  as  the  faulty  length  of 
a  linear  scale,  due  to  its  expansion  from  heat. 


192  THEORY  OF  MEASUREMENTS  §162 

INSTRUMENTAL:    due  to  faulty  graduation  or  adjustment  of 

an  instrument. 
PERSONAL:  some  persons  have  a  constant  tendency  to  estimate 

the  instant  of  an  occurrence  a  little  too  early;  others  a  little 

too  late. 

ACCIDENTAL: 

INSTRUMENTAL:  due  to  varying  external  influences,  "play" 
of  moving  parts,  inconstant  sensitiveness,  etc. 

PHYSIOLOGICAL:  the  senses  of  sight,  touch,  etc.,  have  a  limit 
of  sensitiveness,  and  this  limit  is  not  always  the  same. 

PSYCHOLOGICAL:  very  likely  the  deductions  about  the  outside 
world  that  result  from  the  effect  of  sense-impressions  on  the 
mind  do  not  correspond  to  the  latter  so  closely  as  to  be 
absolutely  free  from  irregular  variations  when  dealing  with 
minute  quantities. 

MISTAKES: 

MANIPULATIVE  :  doing  the  wrong  thing. 

OBSERVATIONAL:  observing  the  wrong  thing. 

NUMERICAL:  recording  the  wrong  numbers.  (It  is  especially 
important  to  guard  against  focussing  the  attention  so  closely 
on  the  minute  part  of  a  measurement,  e.  g.,  the  estimation 
of  tenths  of  the  smallest  scale-division,  that  a  mistake  is 
made  in  recording  the  figures  that  express  the  larger  part  of  it.) 

162.  Accidental  and  Constant  Errors. — The  difference 
between  constant  errors  and  accidental  errors  can  be 
easily  understood  from  the  analogy  of  shots  fired  at 
a  target.  In  the  diagram  the  average  position  or  center 
of  clustering  has  a  constant  tendency  townward  and  to 
the  right  of  the  bull's-eye,  while  the  individual  shots 
have  accidental  tendencies  which  carry  them  to  one  side 
of  this  average  position  as  much  as  to  the  other.  Any 
single  shot  can  be  considered  to  have  a  total  error  which 
is  the  " vector  sum"  or  geometrical  combination  of  its 
own  accidental  error  plus  the  constant  error  of  the  whole 
group.  Physical  measurements  are  target  shots  in  which 
the  center  of  clustering  can  be  found  but  in  which  the 
position  of  the  bull's-eye  is  unknown. 


XV 


MEASUREMENTS  AND  ERRORS 


193 


Constant  errors  can  be  avoided  or  reduced  to  a  mini- 
mum by  the  help  of  theoretical  knowledge  (effect  of 
gravity  in  deflecting  the  shots  downward),  and  by  chang- 
ing observers,  methods,  and  conditions  (repeating  the 
shots  when  the  wind  blows  to  the  left  instead  of  to  the 
right),  and  above  all  by  judgment, 

experience,  care,  and  alert  hunting     t    jfs$*      •  *  • 

for  all  possible  sources  of  error.  ^^\    •  *  *• 

Accidental  errors  are  much  more  .  9  •*•  *  * 
easily  investigated,  the  chief  prob-  •  •  • 

lems  from  the  standpoint  of  phys- 
ical science  being  the  determination 
of  the  center  of  the  cluster  and  the 
measurement  of  the  degree  of  scat- 
tering which  takes  place  around  it. 

State  one  source  of  error  that  was 
present  when  you  performed  each 
of  the  experiments  mentioned  in 
the  following  list.  For  example,  in 
(e),  one  source  of  error  would  be 
the  fact  that  the  periphery  of  the 
disc  is  not  an  exact  circle.  This 
would  give  rise  to  a  constant  error 
in  the  determination  of  ?r  if  the 

diameter  should  be  measured  each  time  along  one  marked 
radius,  but  would  cause  an  accidental  error  if  many  dif- 
ferent diameters  should  be  measured.  Mark  them  in 
such  a  way  as  to  show  which  ones  gave  rise  to  constant 
errors,  and  which  ones  caused  accidental  errors.  Were 
there  any  mistakes? 

(a)  The  measurement  of  your  span. 

(6)  The  measurement  of  an  irregular  area,  first  method. 

(c)  The  measurement  of  the  density  of  an  irregular  solid. 

(d)  The  measurement  of  the  sine  of  an  angle  that  has  a  given 
tangent. 

14 


FIG.  50.  ILLUSTRA- 
TION OF  ERRORS  OF 
MEASUREMENT.  —  The 
shots  (measurements) 
aimed  at  the  bull's-eye 
(true  value)  show  a 
general  drift  (constant 
error)  to  the  right  and 
downward,  and  indi- 
vidual deviations  (acci- 
dental errors)  that  tend 
to  extend  equally  in  all 
directions  from  the  cen- 
ter about  which  they 
cluster. 


194 


THEORY  OF  MEASUREMENTS 


§164 


(e)  The  experimental  determination  of  TT. 
(/)  The  calculation  of  e~*2  for  different  values  of  x. 
(g)  The  use  of  the  formula  1/(1  +  5)  =  1  -  5. 
(h)  The  "black-thread"  determination. 

163.  Errors  and  Variations. — The  theory  of  accidental 
errors  runs  closely  parallel  with  the  theory  of  variation 
in  natural  objects,   so  that  a  statistical   investigation 
of  the  properties  of  several  objects  of  the  same  kind, 
or  variates  as  they  are  technically  called,  will  be  found 
to  illustrate  many  of  the  facts  of  accidental  variation 
which  could  otherwise  be  observed  only  by  the  more 
tedious  study  of  the  deviations  of  repeated  measure- 
ments of  the  same  object. 

164.  Measurement  of  Variates. — Examine  the  vernier 
caliper,  and  if  it  has  an  extra  scale  that  reads  backwards, 
or  one  scale  for  internal  measure- 
ments and  another  for  external 
ones   decide  which   scale   reads 
the    internal    distance   between 
the    jaws    of    the    caliper,    and 
which  point  marks  its  zero  when 
the  jaws  are  closed.     See  that 
you  understand  the  vernier  and 
have  no  trouble  in  reading  it  at 
any  setting. 

Measure  the  length  of  100 
seeds  or  other  variates  of  the 
same  kind.  Make  a  table  of 
preliminary  measurements  of  ten 
variates  in  the  order  in  which 
they  were  measured  to  find  their 
general  range,  and  then  make  a 
table  of  lengths  arranged  in 


cm. 


1.56 

1.58 
1.59 
1.60 
1.61 
1.62 
1.63 
1.64 
1.65 
1.66 
1.67 
1.68 


frequency 


EXAMPLE  OF  A  FRE- 
QUENCY DISTRIBUTION. — 
Notice  that  the  expected 
lengths  are  tabulated  in 
numerical  order  before  the 
actual  measurements  are 
made. 


numerical    order  likethe  above,  recording  each  different 


XV 


MEASUREMENTS  AND  ERRORS 


195 


length  and  the  number  of  times  that  it  occurs.  Do  not 
merely  write  the  length  when  one  seed  is  measured  and 
add  a  pen-stroke  when  another  of  the  same  length  is 
found,  but  see  that  each  variate  is  represented  by  a 
mark  in  the  second  column  of  your  table.  If  an  unex- 
pectedly large  or  small  value  is  found  after  the  first  col- 
umn has  been  written  down  it  may  be  noted  anywhere 
at  the  beginning  or  end  of  the  table,  as  shown  here. 

Plot  a  graphic  diagram  in  which  the  first  ten  variates 
have  their  measurements  in  centimetres  laid  off  along  the 
x-axis  and  the  frequency  of  each  length  is  represented 
by  the  height  of  a  corresponding  ordinate.  Do  not 
begin  the  scale  of  ^-values  at  zero,  but  allow  one  square 
for  each  hundredth  of  a  centimetre.  The  t/-scale  in  any 
statistical  diagram  should  measure  one  square  for  each 
unit  of  frequency,  since  there  are  rarely  enough  measure- 
ments to  require  a  condensed  scale  and  nothing  would 
be  gained  by  an  expanded  scale  where  there  are  no  frac- 
tional values  to  be  plotted.  Connect  the  points  of  the 


Y 

Y 

31 

4 

/ 

\ 

3 

—  I 

/ 

\ 

1 

\ 

2 

6 

x\ 

t 

0 

X 

< 

' 

i 

2 

3 

4 

5 

FIG.  51.  FREQUENCY  POLYGON  AND  HISTOGRAM. — The  ordinary 
graphic  diagram  made  with  broken  lines  is  called  a  frequency  polygon 
if  the  ordinates  represent  the  frequency  of  occurrence  of  the  abscissae. 
When  a  length  along  the  x-axis  is  used  instead  of  a  point,  and  a 
vertical  strip  instead  of  a  line,  the  diagram  is  called  a  histogram. 
The  diagrams  show  two  slightly  different  ways  of  representing 

.  ,        tx  =  0,  1,  2,  3,  4,  5.      AT  ., 

precisely   the   same   facts,   namely,   -<      _         '    '    '    '          Notice 

{.y  —  u,  z,  i,  6,  A,  u. 

particularly  the  difference  in  the  placing  of  the  scale  numbers. 


196  THEORY  OF  MEASUREMENTS  §165 

diagram  by  a  broken  line  and  notice  that  the  " curve" 
is  highest  in  the  middle  and  slopes  downward,  more  or  less 
uniformly,  toward  the  ends.  A  graph  of  this  sort  is 
called  a  frequency  polygon. 

Make  another  graphic  diagram  from  the  same  table, 
but  instead  of  representing  successive  ordinates  by  the 
successive  vertical  ruled  lines  of  the  cross-section  paper 
let  them  be  represented  by  the  successive  white  strips 
that  lie  between  the  vertical  lines  (Fig.  51),  and  use  a 
vertical  scale  of  one  square  for  each  measurement  as 
before.  A  graph  constructed  in  this  way  is  called  a 
histogram.  Notice  that  the  area  inclosed  is  numerically 
equal  to  the  entire  number  of  measurements  or  other 
statistics  that  are  represented. 

Make  a  histogram  from  your  table  of  the  measure- 
ments of  100  variates. 

165.  Questions  and  Exercises. — 1.  In  addition  to  the 
sources  of  error  that  you  have  stated  for  the  experiments 
listed  at  the  end  of  §  162,  write  down  at  least  ten  more 
sources  of  error  that  were  present  when  you  performed 
those  experiments. 

2.  Which  form  of  frequency  diagram  do  you  consider 
to  be  the  more  ''graphic,"  the  histogram  or  the  frequency 
polygon? 


XVI.     STATISTICAL   METHODS 


Apparatus. — Ruler;  slide  rule. 

166.  Frequency  Distributions. — A  tabulation  of  a  set 
of  measurements  that  shows  how  many  times  each 
observed  value  occurs  is  said  to  give  the  frequency  dis- 
tribution of  the  measurements,  and  a  graphic  diagram  in 
which  the  ordinates  give  the  frequencies  of  the  measure- 
ments that  are  represented  by  the  abscissae  is  called  a 
frequency  polygon  if  drawn  with  the  usual  broken  line 
or  a  histogram  if  constructed  by  building  up  rectangles 
on  successive  segments  of  the  base  line. 

Examine  Fig.  52  and  test  it  with  a  ruler  held  hori- 


n  n 


156 


too 


1.68 


f.56 


1.60 


164 


f.68 


FIG.  52.     EXAMPLE  OF  FREQUENCY  DIAGRAMS. — These  correspond 
to  the  table  in  §  164. 

zontally  in  order  to  determine  whether  both  frequency 
diagrams  represent  the  same  frequency  distribution. 
Then  compare,  without  measuring,  the  apparent  heights 
of  the  successive  points  on  the  frequency  polygon  with 
the  successive  values  given  in  the  table  in  §  164.  Has 
any  mistake  been  made  in  attempting  to  represent  that 
table  graphically  by  the  construction  of  Fig.  52? 

197 


198  THEORY  OF  MEASUREMENTS  §167 

167.  Class  Interval. — When  a  set  of  variates  was  meas- 
ured according  to  §  164  the  determinations  were  rounded 
off  to  the  nearest  hundredth  of  a  centimetre  by  the 
inability  of  the  apparatus  to  measure  them  more  accu- 
rately. If  the  five  seeds,  for  example,  which  measure 
1.62  cm.  in  the  printed  table  could  have  been  measured 
with  a  much  higher  degree  of  accuracy  there  can  be  no 
question  that  each  of  them  would  have  a  different  length 
from  any  of  the  others. 

Under  such  circumstances  the  ordinary  type  of  fre- 
quency diagram  could  never  be  obtained,  for  the  fre- 
quencies would  invariably  be  something  like  the  series 
0,  1,  0,  0,  1,  0,  1,  0,  1,  1,  1,  0,  1,  0,  1,  1,  0,  1,  1,  0,  1,  0,  1,  0, 
and  the  diagram  would  no  longer  have  its  characteristic 
"mound-like"  shape  with  the  ordinates  high  in  the 
middle  of  the  graph  and  dwindling  away  toward  each  end. 

The  shape  of  a  frequency  diagram,  accordingly,  may  de- 
pend upon  the  numerical  distance  between  the  successive 
abscissse  that  are  used,  the  least  difference  between  re- 
corded measurements.  In  the  table  of  §  164  this  amounts 
to  one  hundredth  of  a  centimetre,  but  if  those  lengths 
had  been  rounded  off  to  the  nearest  half  millimetre 
the  frequency  distribution  would  have  become 

f  1.55         1.60         1.65         1.70  centimetres, 
[    1  9  41  2    cases, 

in  which  little  is  left  of  the  original  shape  of  the  graph 
except  a  suggestion  of  the  general  " mound-like"  char- 
acter. 

The  least  difference  in  successive  recorded  measure- 
ments is  .05  cm.  in  the  last  illustration,  .01  cm.  in  the 
original  table  and  Fig.  52,  and  is  called  the  class  interval. 

If  your  frequency  polygon  tends  to  the  nondescript 
type  seen  when  the  class  interval  is  too  small  the  meas- 


XVI  STATISTICAL  METHODS  199 

urements  of  the  hundred  variates  should  be  regrouped. 
Make  a  new  table  in  which  the  values  from  1.55  to  1.65 
are  all  called  1.6,  those  from  1.65  to  1.75  are  considered 
as  1.7,  etc.  If  several  measurements  have  been  recorded 
as  1.65  put  about  half  of  them  in  the  class  1.6  and  the 
other  half  in  the  class  1.7.  From  this  table  plot  sepa- 
rately both  the  frequency-polygon  and  the  histogram 
that  illustrate  it  graphically.*  Do  not  use  an  expanded 
scale  in  any  statistical  graph  either  for  frequency  or  for 
class  interval. 

168.  Types  of  Frequency  Distribution. — If  the  class 
interval  is  made  very  small  and  the  measurements  are 
very  numerous,  both  forms  of  diagram  can  be  considered 
as  losing  their  abrupt  changes  until  they  merge  into 
two  identical  curved  lines,  the  notches  that  were  orig- 
inally present  having  become  indefinitely  small.  Fre- 
quency distributions  are  not  invariably  " mound-shaped," 
but  may  be  classified,  according  to  the  general  shape  of 
the  graphic  diagram,  as  (a)  symmetrical,  (b)  moderately 
asymmetrical,  (c)  very  asymmetrical  or  J-shaped,  (d) 
bilocular  or  U-shaped,  (e)  rectangular  (Fig.  53). 

The  symmetrical  type  (a)  is  seen  in  physical  meas- 
urements. Notice  that  in  it  (1)  the  average  (i.  e.} 
middle)  z- value  is  the  most  frequent;  (2)  measurements 
that  are  a  given  amount  above  or  below  the  average  are 
less  frequent  than  it  but  of  equal  frequency  with  each 
other;  and  (3)  extremely  divergent  values  do  not  occur. 
The  asymmetrical  type  (6)  is  similar  except  that  the 
most  frequent  value  does  not  lie  in  the  middle  of  the 
distribution  and  on  one  side  of  it  the  frequency  falls 
away  more  rapidly  than  on  the  other.  Moderate  asym- 
metry is  common  in  all  statistical  data.  An  example  of 

*  If  preferred,  a  class  interval  of  .05  cm.  may  be  used  instead. 
This  will  avoid  splitting  any  original  class  into  two. 


200 


THEORY  OF  MEASUREMENTS 


the  J-shaped  type  (c)  may  be  seen  in  the  distribution  of 
wealth  among  any  population;  the  frequency  of  indi- 
viduals with  little  wealth  is  very  great,  and  with  in- 


/    b 


a 


FIG.  53.  TYPES  OF  FREQUENCY  DISTRIBUTION. — (a)  Symmet- 
rical. (6)  Positively  asymmetrical  (skewed  toward  the  right),  (c) 
Extreme  asymmetrical,  or  J-shaped.  (d)  Bilocular,  or  U-shaped. 
(e)  Rectangular. 

creasing  wealth  the  number  of  cases  falls  off  until  it- 
reaches  a  vanishing  point.  The  curious  U-shaped  type 
(d)  is  seen  where  there  are  tendencies  toward  both 
extremes,  or  the  centre  is  a  position  of  unstable  equilib- 
rium; and  the  rectangular  type  (e)  occurs  in  purely 
mathematical  cases,  such  as  the  actual  error  of  a  tabu- 
lated logarithm,  which  is  never  more  than  d=  0.5  of  the 
unit  in  the  last  decimal  place  and  has  all  intermediate 
values  with  equal  frequency. 

169.  The  Probability  Curve. — The  measurement  of  100 
seeds  will  probably  show  a  moderate  degree  of  asym- 


XVI  STATISTICAL  METHODS  201 

metry,  since  such  objects  do  not  grow  beyond  a  definite 
size  but  do  fall  short  of  it  in  many  cases.  This  is  called 
negative  asymmetry,  and  the  curve  is  said  to  be  skewed 
toward  the  left.  Physical  measurements,  however,  tend  to 
be  above  the  average  just  as  often  as  they  are  below  it,  and 
so  give  the  symmetrical  form  of  frequency  distribution. 
As  the  measurements  are  made  more  and  more  numerous 
the  frequency  polygon  approaches  more  and  more  closely 
to  the  form  of  the  so-called  probability  curve,  y  =  e~**, 
which  was  calculated  in  the  lesson  on  logarithms  and 
plotted  in  the  lesson  on  graphic  representation.  In 
some  cases  it  will  appear  drawn  out  relatively  flat  and 
in  others  will  be  very  high  and  narrow  (Fig.  34),  but  the 
curve  is  the  same  in  all  cases,  except  that  the  scales  of 
^-values  and  ^/-values  are  condensed  or  spread  out  to 
different  degrees. 

Carry  out  the  binomial  expansion  that  is  given  below 
at  least  as  far  as  n  =  15,  noticing  that  each  two  succes- 
sive terms  have  to  be  added  in  order  to  obtain  the  term 
below  and  between  them.  Lay  off  a  series  of  equal 
intervals  on  the  z-axis.  On  these  erect  a  series  of  ordi- 
nates  proportional  to  the  successive  terms  of  the  last 
polynomial,  in  order.  A  smooth  curve  through  the  tops 
of  the  ordinates  will  give  a  very  good  approximation 
to  the  probability  curve. 

(1  +  D°  =  1 

(1  +  I)1  =  1  +  1 

(1  +  I)2  =  1+2  +  1 

(1  +  1)3  =  1+3+3  +  1 

(1  +  I)4  =  1+4  +  6  +  4  +  1 
(1  +  I)5  =1+5  +  10  +  10  +  5  +  1 
(1  +  1)"  = 

170.  Representative  Magnitudes. — For  most  scientific 
work,  the  statement  of  a  whole  frequency  distribution 


202  THEORY  OF  MEASUREMENTS  §171 

or  of  each  one  of  a  long  series  of  measurements  would 
be  a  cumbrous  process  and  reading  such  statements 
would  be  a  tedious  task.  Accordingly  it  is  customary 
to  summarize  such  a  set  of  values  by  stating  some 
representative  value,  such  as  the  average.  In  special 
cases  such  representative  magnitudes  as  the  geomet- 
rical mean,  M  (oiO2-  •  -an),  or  the  harmonic  mean, 
n/(l/di  +  1/02  +  •  •  •  +  I/On),  or  the  quadratic  mean, 
V  [(«i2  +  o22  +  •  •  •  +  on2) /n],  have  been  employed,  but 
the  most  usual  one  is  the  arithmetical  mean  or  average, 
(01  +  o2  +  •  •  •  H-  an)/n.  The  median,  which  is  on  if 
01,  o2  •  •  •  o2tt_i  are  arranged  in  order  of  size,  is  frequently 
useful  as  a  representative  value,  as  is  also  the  mode 
or  modal  value,  which  is  simply  the  value  that  occurs 
with  the  greatest  frequency.  The  word  mean  is  some- 
times used  in  a  general  sense  for  any  representative  value, 
and  sometimes  is  restricted  to  the  same  significance  as 
the  word  average. 

171.  The  Average — The  average  is  obtained  by  adding 
together  a  set  of  values  and  dividing  the  sum  by  the 
number  of  values.     Thus  the  average  of  the  five  values 
3,  3,  4,  5,  10,  is  one  fifth  of  their  sum,  or  5.     If  certain 
values  occur  repeatedly  the  average  is  obtained  by  divid- 
ing the  sum  by  the  number  of  values,  not  by  the  number 
of  different  values.     Thus,  the  average  of  the  measure- 
ments given  in  the  table  at  the  end  of  the  preceding 
lesson  is  (2  X  1.68  +  6  X  1.67  +  3  X  1.66  -  •  -)/(2  +  6 
+  3  -•-),  or  8686/53  or  1.639. 

172.  The  Median. — The  median  is  obtained  by  choos- 
ing such  a  value  that  half  of  the  other  values  exceed  it 
and  half  are  below  it.     //  the  numbers  are  arranged  in 
numerical  order  in  a  column  the  number  that  is  half  way 
down  the  column  is  the  median.     Otherwise  it  may  be 
found  by  crossing  off  the  largest  number  and  the  smallest, 


XVI  STATISTICAL  METHODS  203 

and  repeating  the  process  until  only  one  is  left.  If  two 
numbers  are  left,  as  will  be  the  case  if  there  are  an  even 
number  of  measurements,  the  number  half  way  between 
them  can  be  taken  as  the  median,  but  in  physical  or 
statistical  data  it  usually  happens  that  the  two  remaining 
numbers  are  the  same. 

Find  the  median  of  3,  3,  4,  5,  10.  What  is  the  median 
of  7.4,  6.8,  7.3,  7.3,  7.2,  7.1,  7.2?  Ans.:  7.2.  Find  the 
median  of  the  measurements  that  are  tabulated  in  §  164. 

Turn  to  the  histogram  in  Fig.  51,  §  164,  and  cross  off 
one  of  the  small  squares  between  the  curve  and  the  base 
line  at  the  extreme  left  and  then  one  at  the  extreme  right ; 
repeat  the  process  until  only  one  (or  two)  squares  are 
left,  and  show  that  the  median  is  3.  Imagine  that  Fig. 
54  is  similarly  divided  up  into  small  squares,  each  square 
representing  one  measurement,  and  satisfy  yourself  of 
the  truth  of  the  fact  that  the  ordinate  that  is  drawn  up- 
ward from  the  median  x-value  must  bisect  the  area  of  the 
frequency  curve. 

173.  The  Mode. — The    mode    is    the    most    frequent 
value.     Thus  in  the  set  of  values  3,  3,  4,  5,  10  the  mode 
is  the  number  that  occurs  twice,   namely  3.     In  the 
illustration  of  the  measurement  of  variates  the  mode  is 
1.64,  the  length  that  was  found  most  often. 

174.  Choice  of  Means. — If  a  frequency  distribution  is 
of  the  symmetrical  type  the  average,  mode,  and  median 
will  all  be  the  same. 

Where  there  is  a  moderate  degree  of  asymmetry  the 
median  will  come  nearer  to  the  mode  than  the  aver- 
age does,  as  is  shown  in  the  following  diagram.  The 
mode  is  easily  seen  to  be  the  most  probable  value.  If 
a  seed  is  taken  at  random  from  the  set  whose  measure- 
ments were  given  in  the  table  it  is  more  likely  to  measure 
1.64  cm.  than  any  other  amount.  The  mode  has  one 


204 


THEORY  OF  MEASUREMENTS 


§174 


decided  disadvantage,  however,  in  that  it  cannot  always 
be  chosen  from  a  given  frequency  distribution.  For 
example  the  mode  cannot  be  obtained  from  the  series 


Qf        Mo  Me  Av  Q3 

FIG.  54.  TYPICAL  FREQUENCY  DIAGRAM. — The  mode  has  the 
highest  ordinate.  The  ordinate  of  the  median  bisects  the  area  under 
the  curve.  The  ordinate  of  the  average  passes  through  the  center 
of  gravity  of  the  area.  The  median  lies  between  the  mode  and  the 
average,  and  is  twice  as  far  from  the  former  as  from  the  latter.  The 
difference  (distance)  between  the  quartiles  Q\  and  Qs  (§  178)  is 
called  the  interquartile  range. 

3,  4,  4,  5,  6,  6,  7,  8,  except  by  assuming  that  the  values 
follow  the  probability  law,  and  fitting  a  probability 
curve  to  them  as  accurately  as  possible  by  a  "  black 
thread"  method;  the  position  of  the  top  of  this  theoret- 
ical curve  can  then  be  determined.  Where  there  is  a 
long  series  of  measurements  the  average  is  sometimes 
calculated  from  the  formula,  indicated  on  the  above 
diagram,  that  me  —  mo  =  2  (av  —  me),  or  that  the 
median  lies  $  of  the  way  from  the  average  to  the  mode. 

The  median,  like  the  mode,  is  easy  to  determine. 
It  can  be  used  in  two  classes  of  cases  where  the  average 
cannot  be  determined.  One  is  in  case  measurements 
have  been  tabulated  with  indefinite  terminal  classes, 
e.  g.,  "less  than  10  mm.,  7  cases;  10  to  12  mm.,  4  cases; 

12  to  14,  5;   14  to  16,  3;   above  16,  2  cases."     Here  the 
median  is  the  class  "between  10  and  12,"  say  11  mm., 
and  the  mode  is  probably  the  class  "12  to   14,"  say 

13  mm.,  but  the  average  cannot  be  found  on  account 


XVI  STATISTICAL  METHODS  205 

of  nine  of  the  numerical  values  not  being  stated;  the 
best  that  could  be  done  would  be  to  guess  that  the 
average  was  a  little  less  than  the  median.  The  other 
case  is  where  quantities  can  be  arranged  in  numerical 
order  but  are  difficult  to  measure  individually.  Thus 
it  may  be  difficult  or  impossible  to  gauge  the  scholar- 
ship of  a  student  in  accurate  numerical  terms,  but  if  a 
group  of  students  can  be  arranged  in  order  of  scholar- 
ship the  median  can  be  determined  without  difficulty. 
Furthermore  the  median  has  a  unique  advantage  over 
most  other  representative  magnitudes  in  that  it  is  not 
affected  by  inverting  the  unit  of  measurement.  The 
median  of  a  number  of  prices  will  be  the  same  whether 
they  are  given  as  cents  per  dozen  or  as  dozens  per  dollar. 
Of  a  group  of  different  velocities  the  same  one  will  be 
picked  out  by  choosing  the  median  whether  the  numer- 
ical values  are  expressed  in  miles  per  hour  or  in  minutes 
for  a  one-mile  run.  It  is  easy  to  see  that  this  will  not  be 
the  case  if  the  average  is  used.  The  median,  however,  is 
not  quite,  as  good  a  representative  value  as  the  average, 
in  case  they  are  different,  for  a  series  of  measurements 
that  have  all  been  made  with  equal  care.  Curiously 
enough,  however,  the  more  extensive  a  series  of  meas- 
urements the  more  likely  it  is  to  show  that  the  individual 
measurements  are  not  equally  trustworthy  but  may 
be  grouped  in  different  classes  according  to  their  rela- 
tive scattering.  It  is  in  such  cases  that  the  median  is 
a  much  better  representative  figure  than  the  average, 
for  the  average  is  influenced  by  an  unduly  large  or  small 
measurement  just  in  proportion  to  the  aberration  of 
the  latter,  while  as  long  as  a  measurement  is  above  the 
median  it  makes  no  difference  how  far  above  it  may  be, 
its  effect  is  no  greater  than  that  of  any  other  single 
value.  (Compare  the  average  and  the  median  of  3,  3, 


206  THEORY  OF  MEASUREMENTS  §175 

4,  5,  1O,  with  those  of  3,  3,  4,  5,  30.  The  very  erratic 
values  ought  to  have  the  least  influence  instead  of  the 
most.) 

175.  Deviations. — Of  almost  as  much  importance  as 
finding  the  best  representative  value  for  a  series  of 
measurements  is  the  determination  of  how  closely  they 
cluster  around  it  or  how  widely  they  scatter  from  it. 
This  will  help  to  furnish  information  in  regard  to  the 
accuracy  of  the  measurements,  and  hence  also  in  regard 
to  the  accuracy  of  the  instruments  and  methods  em- 
ployed in  making  them;  it  will  also  be  useful  in  com- 
paring and  combining  determinations  made  at  differ- 
ent times  or  by  different  observers.  The  difference, 
m  —  a,  between  any  single  measurement  and  the  average 
(or  other  representative  magnitude)  is  known  as  the 
deviation  or  variation  of  that  measurement  and  is  denoted 
by  the  letter  v.  It  is  not  the  same  as  the  error  of  the 
measurement,  for  the  error  may  have  a  constant  com- 
ponent that  affects  all  of  the  measurements  equally; 
but  it  may  be  considered  as  the  accidental  error  or 
accidental  component  of  the  total  error.  Deviations 
give  no  positive  indication  of  any  constant  errors  that 
may  be  present. 

If  each  one  of  a  series  of  independent  measurements 
is  as  trustworthy  as  any  other  it  can  be  shown  mathe- 
matically that  their  best  representative  value  is  their 
arithmetical  mean,  or  average;  and  accordingly  the 
average  is  the  figure  that  is  almost  invariably  used. 

Copy  the  two  following  columns  of  figures,  find  the 
average  of  each  set,  and  write  after  each  measurement 
(m)  its  deviation  (v)  from  the  average  of  the  figures 
in  the  same  column,  marking  it  with  a  minus  sign  if  the 
value  is  less  than  the  average,  and  with  a  plus  sign  if  in 
excess  of  the  average. 


XVI 


STATISTICAL  METHODS 


207 


m 

V 

m 

V 

27.35 

27.36 

27.34 

27.38 

27.34 

27.35 

27.33 

27.37 

27.34 

27.32 

27.34 

27.30 

27.33 

27.31 

27.34 

27.30 

27.35 

27.35 

27.34 

— 

27.36 

—  • 

Although  both  averages  are  the  same  it  is  evident 
that  the  first  set  of  measurements  must  have  been  made 
by  a  more  trustworthy  instrument,  observer,  or  method 
than  the  second.      If  the  aver- 
ages had  not  been  of  the  same 
value  the   first  one  would  un- 
doubtedly have  been  entitled  to 
more    confidence,    other    things 
being  equal,  than  the  second. 

Add  each  column  of  deviations 
and  verify  the  fact  that  the  alge- 
braical sum  of  the  deviations  from 
the  average  is  always  zero.  If 
their  sum  is  zero  their  average 
will  also  be  zero,  so  that  when  an  " average  deviation" 
is  recorded  the  term  means  not  the  average  of  the  alge- 
braical deviations  but  the  average  of  the  values  that 
they  would  have  if  the  negative  signs  were  omitted. 

176.  Average  by  Symmetry. — The  foregoing  property 
suggests  an  easy  method  of  finding  the  average  in  simple 
cases:  If  a  number  can  be  so  chosen  that  the  individual 
measurements  are  symmetrically  grouped  around  it  the 
sum  of  the  positive  deviations  will  equal  the  sum  of  the 
negative  deviations  and  the  number  will  be  the  required 
average. 

What  is  the  average  of  115  and  119?  Ans.:  117, 
because  it  makes  the  sum  of  the  deviations  (+2  and 
—  2)  equal  to  zero. 

Find  the  average  of  3,  6,  9,  12,  15  by  the  method  of 
symmetry. 

What  is  the  average  of  16,  18,  20,  22?  Of  14  and  17? 
Of  12,  14,  17,  19?  Of  126.8  and  127.4?  Of  121  and 
141?  Of  161  and  191?  Of  198,  199,  203?  Of  8,  10J, 
11 1?  What  is  the  value,  to  the  nearest  whole  number, 


208  THEORY  OF  MEASUREMENTS  §178 

of  the  average  of  117,  116,  117?  (Suggestion:  Is  the 
average  above  or  below  116.5?)  What  is  the  exact 
average  of  17,  18,  18,  19,  19,  20? 

177.  Average  by  Partition. — Before  leaving  the  sub- 
ject of  the  average  it  should  be  noted  that  there  is  no 
need  of  adding  the  entire  numerical  values  if  they  con- 
tain a  part  in  common.     Thus  in  either  of  the  columns 
of  figures  in  §  175  the  average  is  obtained  by  writing 
the  first  three  figures,  which  are  common  to  all  values, 
and  annexing  the  average  of  the  last  figure. 

178.  Quartiles. — Just  as  the  middle  number  of  a  series 
arranged  in  ascending  or  descending  order  of  magnitude 
is  called  the  median,  or  half-way  value,  so  the  middle 
one  of  the  numbers  that  lie  below  the  median  is  called 
the  lower  quartile,  or  quarter- way  value;  and  the  median 
of  the  numbers  that  lie  above  the  median  is  known  as 
the   upper   quartile,    or   three-quarter-way   value.     The 
quartile  abscissae  are  laid  off  on  the  base  line  of  Fig.  54, 
§  174,  at  Qi  and  Q3.     It  should  be  carefully  noticed  that 
their  ordinates,  together  with  the  median  ordinate,  divide 
the  whole  area  of  the  frequency  diagram  into  four  equal 
parts.     If  there  is  any  difficulty  in  understanding  this 
remember  that  the  area  of  a  histogram  indicates  the 
total  number  of  measurements  (§  164).     If  half  of  them 
lie  below   the   median,   by   definition,   and   half  above 
there  can  be  no  trouble  in  realizing  that  the  median  is  the 
abscissa  whose  ordinate  bisects  the  area  of  the  figure. 
Of  course  the  same  thing  is  not  true  of  the  average, 
except  when  average  and  median  happen  to  have  the 
same  value. 

What  are  the  quartiles  of  the  series  3,  4,  4,  5,  6,  8,  11? 

Find  the  median  and  quartile  values  of  the  set  of 
numbers  28,  29,  31,  36,  31,  30,  35,  33,  32,  36,  29,  28,  32. 
Notice  that  the  numbers  " below"  the  median  need  not 


XVI  STATISTICAL  METHODS  209 

all  be  smaller  numbers;  the  median  is  31  and  one  of  the 
lower  numbers  is  also  31.  Notice  that  the  lower  quartile 
is  the  median  of  the  six  numbers  below  the  median, 
not  of  the  seven  numbers  that  include  the  median  itself. 
Similarly,  the  upper  quartile  is  not  33,  but  34  (§  172). 

179.  Semi-Interquartile  Range. — The  numerical  dis- 
tance between  the  lower  quartile  and  the  upper  quartile 
is  called  the  interquartile  range.  In  the  series  7,  8,  10, 
13,  19,  22,  27,  the  quartiles  are  8,  and  22,  and  their 
difference,  14,  is  the  range  between  quartiles,  or  inter- 
quartile range.  Half  of  this,  or  7,  is  called  the  semi- 
interquartile  range;  notice  that  it  is  the  average  of  the 
two  distances  from  median  to  quartile.  It  is  sometimes 
used  as  a  measure  of  the  scattering  or  clustering  of  a 
set  of  observations,  and  its  theoretical  importance  will 
be  seen  later. 

Find  the  semi-interquartile  range  of  each  of  the  two 
columns  of  figures  in  §  175  after  re-arranging  the  data 
in  numerical  order.  For  the  first  column:  if  the  five 
smaller  values  are  considered  as  lying  below  the  median 
and  five  above,  that  is,  if  the  median  is  considered  to 
occupy  no  numbers  in  the  middle  of  the  column  the 
lower  quartile  will  be  27.34;  but  if  only  four  values  are 
considered  as  lying  below  the  median,  and  four  above, 
that  is,  if  the  median  is  considered  to  include  two  numbers 
in  the  middle  of  the  column  the  lower  quartile  will  be 
27.335.  As  the  median  is  actually  considered  to  be 
neither  two  numbers  nor  no  numbers,  but  is  one  single 
number  it  will  be  satisfactory  to  say  that  the  lower  quar- 
tile is  neither  27.340  nor  27.335,  but  is  half  way  between 
these  two  values,  say  27.338.  Similarly,  show  that  the 
upper  quartile  is  27.342.  The  second  column  should  be 
treated  in  the  same  manner.  What  do  you  notice  about 
the  relative  size  of  the  two  semi-interquartile  ranges? 
15 


210  THEORY  OF  MEASUREMENTS  §180 

180.  Questions  and  Exercises. — 1.  Complete  the  fol- 
lowing rule  for  finding  the  upper  quartile  of  12  measure- 
ments, and  submit  it  to  your  instructor  for  inspection: 
"  Arrange  the  measurements  in  order  of  magnitude  and 
find  the  median  by  taking  the  average  of  the  sixth  and 
seventh.  Then  .  .  ." 

2.  When  the  tabulated  values  of  §  164  are  re-grouped 
in  larger  class  intervals  (§  167)  there  are  seven  measure- 
ments of  1.65  cm.,  which  theoretically  should  be  sepa- 
rated into  two  equal  groups  for  inclusion  with  the  classes 
1.60  cm.  and  1.70  cm.     Would  it  be  objectionable  to  add 
exactly  3  J  squares  to  each  of  the  strips  of  the  histogram,  or 
would  it  be  better  to  add  3  to  one  and  4  to  the  other? 
Explain  why. 

3.  Why  does  it  generally  happen  that  the  median  of 
a  set  of  measurements  fails  to  be  exactly  twice  as  far 
from  the  mode  as  from  the  average? 

4.  Draw  roughly  a  curve  like  Fig.   54,   but  having 
negative    instead    of    positive    asymmetry.     Place    the 
mode  and  the  median  where  you  think  they  should  go, 
and  then  locate  the  average. 

5.  Make  a  tabular  statement  of  (a)  the  advantages 
of  the  median  over  the  average,  and  (6)  the  advantages 
of  the  average  over  the  median,  as  a  representative  value. 

6.  Prove  that  for  any  three  numbers  (say,  k  +  di,  etc.) 
the  algebraical  sum  of  the  deviations  from  the  average 
must  be  equal  to  zero. 

7.  If  a  train  travels  the  first,  third,  fifth,  seventh, 
.  .  .  miles  at  a  rate  of  20  miles  per  hour,  and  the  second, 
fourth,  sixth,  eighth,  .  .  .  miles  at  30  miles  per  hour, 
show  that  its  average  velocity  is  not  the  average  of  the 
two  rates,   25  miles  per  hour,   but  is  their  harmonic 
mean  (§  170)  instead. 

8.  What  would  you  suggest  for  a  numerical  measure  of 
asymmetry  in  a  diagram  like  Fig.  54? 


XVII.     DEVIATION  AND   DISPERSION 

Apparatus. — Vernier  caliper;  variates;  slide  rule;  table 
of  logarithms. 

181.  Character istic  Deviations. — Just  as  the  use  of  an 
average  or  other  representative  magnitude  obviates  the 
necessity  of  stating  the  separate  measurements  from 
which  it  is  derived,  so  a  statement  of  all  the  deviations 
from  the  average  can  be  replaced  by  a  single  character- 
istic  deviation.     A   set   of   measurements   can   then   be 
summarized  by  two  numbers,  and  it  is  customary  to 
write  such  a  result  in  the  form  a  ±  d*  where  the  first 
number  gives  the  average  value  of  the  quantity  measured 
and  the  second  expresses  the  limiting  distances  above 
and  below  the  average  which  mark  off  in  some  way  an 
amount  of  deviation  that  is  characteristic  for  the  set 
of  measurements. 

182.  Total  Range, — The  simplest  way  of  summarizing 
the  deviations  of  a  series  of  measurements  is  by  stating 
their  total  range,  or  the  algebraical  difference  between 
the  smallest  one  and  the  largest  one.     Obviously  the 
extreme  range  of  the  measurements  themselves  will  also 
give  the  same  result.     Thus,   in  the  two   columns  of 
measurements  in  §  175,  show  that  the  total  range  is 
.02  for  the  first  and  .08  for  the  second.     Unfortunately, 
this  is  not  only  the  easiest  but  also  the  worst  method  of 
obtaining  a  characteristic  deviation,  for  the  numerical 
value  given  by  the  total  range  depends  upon  only  two 
of  the  measurements  of  the  series;    and  those  two  are 

*  The  letter  d  will  be  used  in  this  book  to  denote  a  particular  kind 
of  characteristic  deviation.  A  single  deviation  of  any  individual 
measurement  will  be  indicated  by  the  letter  v  (variation). 

211 


212  THEORY  OF  MEASUREMENTS  §184 

the  least  satisfactory  ones,  because  a  repetition  of  the 
series  of  measurements  would  be  likely  to  show  a  con- 
siderable fluctuation  in  the  largest  and  smallest  values, 
while  the  most  closely  clustered  values  would  hardly  be 
changed  at  all. 

What  is  the  numerical  value  of  the  total  range  of  the 
data  in  the  table  of  §  164?  In  what  kind  of  units  should 
it  be  expressed? 

183.  Average    Deviation. — A    better    index    of    the 
amount  of  scattering  is  given  by  the  average  deviation. 
This  is  the  average  of  the  positive  arithmetical  values 
of  all  of  the  deviations.     The  true  algebraical  average 
of  the  deviations   cannot  be  used  if  they  have  been 
calculated  from  the  average  measurement  because,  as 
has  been  shown  (§  175),  its  value  is  always  zero. 

The  average  of  the  positive  values  of  the  deviations 
is  always  smallest  when  the  deviations  have  been  cal- 
culated from  the  median  measurement,  and  it  is  in  con- 
nection with  the  median  that  the  average  deviation  is 
generally  used. 

Find  the  average  and  the  median  of  the  numbers  3, 
3,  4,  5,  10.  Then  find  the  deviation  of  each  of  these 
numbers  from  the  median,  being  careful  not  to  omit  one 
of  the  deviations  in  case  it  happens  to  be  zero.  What  is 
the  value  of  the  average  deviation  when  calculated  from 
the  median?  Find  also  the  five  deviations  from  the  aver- 
age, and  determine  the  average  of  their  (positive)  arith- 
metical values.  Is  the  average  deviation  also  smaller 
when  measured  from  the  median  than  when  measured 
from  the  mode? 

184.  Standard  Deviation. — The  typical  deviation  that 
is  most  frequently  used  for  statistical  purposes  is  the 
standard  deviation.     This  is  the  square  root  of  the  quo- 
tient of  the  sum  of  the  squares  of  the  deviations  from 


XVII  DEVIATION  AND  DISPERSION  213 

the  average  divided  by  one  less  than  the  number  of 
statistical  values.*  If  there  are  n  quantities  whose 
deviations  from  their  average  are  respectively  Vi,  v2,  v3, 
•  -  •  vn,  the  standard  deviation  of  those  quantities  is  the 
value  of 


n  -  1 

The  standard  deviation  is  a  special  case  of  the  mean- 
square  deviation  (or  root-mean-square  deviation),  being  in 
fact  the  mean-square  deviation  from  the  average,  or 
approximately  the  quadratic  mean  of  the  deviations 
from  the  average.  It  is  also  sometimes  called  the  mean 
deviation  and  the  mean-square  deviation,  but  as  the  term 
mean  deviation  is  also  used  for  what  we  have  defined  as 
the  average  deviation  it  is  much  better  to  avoid  the  use 
of  the  word  mean  (§  170)  altogether  in  connection  with 
a  deviation. 

185.  Dispersion. — The  measure  of  deviation  which  is 
used  for  physical  measurements  in  almost  all  cases  is  that 
particular  form  of  characteristic  deviation  which  is  called 
the  dispersion.  It  is  approximately  two  thirds  of  the 
standard  deviation,  or,  more  exactly, 

±.674490     /*'  +  *'+•••  + 


n  -  1 

186.  Significance  of  the  Dispersion. — An  important 
characteristic  of  this  typical  value  is  that  for  a  series  of 
measurements  which  is  extensive  enough  to  follow  the 

*  To  be  strictly  accurate,  the  standard  deviation  of  the  statis- 
tician is  V  2v2/  V  n.  The  reasons  for  using  n  —  1  will  be  found  in 
the  larger  treatises;  here  it  need  only  be  noticed  that  the  denominator 
is  diminished  because  the  numerator  is  smaller  than  the  sum  of  the 
squares  of  the  true  errors  (§216).  Of  course,  if  n  is  fairly  large, 
the  distinction  is  unnecessary. 


214  THEORY  OF  MEASUREMENTS  §187 

law  of  the  probability  curve  the  dispersion  can  be  shown 
mathematically  to  express  exactly  the  same  limits  above 
and  below  the  average  as  are  given  with  respect  to  the 
median  by  the  semi-interquartile  range.*  If  measurements 
follow  the  law  of  the  probability  curve  the  median  and 
the  average  will  coincide;  and  the  upper  and  lower 
quartiles  will  include  within  the  space  between  them 
just  half  of  the  total  number  of  measurements,  as  was 
seen  in  connection  with  Fig.  54.  In  other  words,  the 
right-hand  half  of  the  frequency  diagram  will  be  bisected 
by  the  ordinate  corresponding  to  the  positive  value  of 
the  dispersion,  and  the  corresponding  abscissa  will  be 
the  median  value  of  all  the  positive  deviations.  Since 
the  left-hand  half  of  the  curve  is  likewise  bisected  by 
the  negative  value  of  the  dispersion,  and  the  graphic 
diagram  is  symmetrical  with  respect  to  the  ?/-axis,  it 
follows  that  the  dispersion  is  the  median  of  the  absolute 
arithmetical  values  of  all  the  deviations,  or  that  any 
single  deviation  is  as  likely  to  be  less  than  the  dispersion  as 
it  is  to  be  greater.  The  dispersion,  with  its  plus-and- 
minus  sign  marks  off  a  small  distance  above  and  below 
the  average,  and  it  is  just  as  likely  as  not  that  any  single 
measurement  chosen  at  random  will  be  within  these 
limits. 

187.  Advantage  of  the  Dispersion. — Where  the  num- 
ber of  measurements  in  a  series  is  relatively  small,  say 
not  greater  than  10,  the  dispersion  obtained  by  calcula- 
tion and  the  semi-interquartile  range  obtained  by  picking 
out  the  quartile  values  will  not  usually  have  exactly  the 
same  value,  on  account  of  the  measurements  not  being 
sufficiently  numerous  to  follow  the  laws  of  probability 

*  Show  that  the  median  and  semi-interquartile  range  of  the  53 
measurements  given  in  the  table  of  §  164  are  1.64  cm.  and  0.01  cm. 
Could  these  two  numbers  be  used  as  a  rough  approximation  for 
average  and  dispersion? 


XVII 


DEVIATION  AND  DISPERSION 


215 


very  closely.  Even  in  this  case,  however,  the  value 
given  by  the  formula  is  preferable  to  half  the  difference 
between  the  quartiles 
because  the  former  is 
obtained  from  all  the 
measurements  of  the 
series  and  the  latter 
from  only  two.  Where 
there  are  as  many  as 
ten  measurements  of  a 
physical  magnitude  it 
is  usually  found  that  the 
two  values  will  not  differ 
by  more  than  15  or  20  per 
cent.,  and  for  rough  pre- 
liminary measurements 
the  semi  -  interquartile 
range  is  almost  as  satis- 
factory as  the  dispersion 
and  can  be  obtained 
much  more  readily. 

188.  Calculation  of  the  Dispersion. — The  table  shows 
the  measurements  (m)  of  ten  variates  taken  at  random 
from  the  100  that  were  measured  before,  also  their 
average  (av),  and  their  individual  deviations  (v)  written 
without  decimal  points.*  The  third  column  shows  the 
squares  of  the  numbers  in  the  second  column,  as  obtained 
from  the  table  of  squares  (see  appendix).  According  to 
the  formula  in  §  185  the  sum  of  these  squares  is  to  be 
divided  by  9,  the  square  root  of  this  quotient  is  then  to 
be  found  and  multiplied  by  about  f  in  order  to  obtain 
the  dispersion  of  this  set  of  measurements. 

*  In  the  calculation  of  a  dispersion  the  deviations  should  always 
be  written  down  as  whole  numbers.  Nothing  would  be  gained  by 
writing  these  v's  in  centimetres  instead  of  in  thousandths  of  -a 
centimetre. 


m 

V 

v2 

1.82  cm. 
1.85 
1.85 
1.78 
1.82 
1.84 
1.78 
1.62 
1.81 
1.70 

33 
63 
63 
7 
33 
53 
7 
.167 
23 
87 

1089 
3969 
3969 
49   * 
1089 
2809 
49 
27889 
529 
7569 

17.87  (10 
1.787.  .av 

49010...  2  (v2) 

log  49010  =  4.6903 
log  9  =  .9542 

2)3.7361 

1.8680 
log  .6745  =  1.8290 

1.6970  =  log  d 
d  =  49.78 


216  THEORY  OF  MEASUREMENTS  §190. 

If  the  numbers  in  column  v  are  expressed  in  thou- 
sandths of  a  centimetre  the  dispersion  will  also  come 
out  in  thousandths;  thus,  the  value  of  d  shown  in  the 
illustration  represents  .04978  cm.  This  means  that  the 
limits  1.737  cm.  and  1.837  cm.  are  the  boundaries  between 
which  about  half  of  the  measurements  should  lie^see  §  186). 
Verify  this  from  the  tabular  values  of  m. 

189.  Rule  for  Accuracy  of  the  Average. — In  determin- 
ing how  many  figures  of  the  average  are  to  be  retained 
as  significant  it  is  best  to  follow  the  rule  that  at  least 
half  of  the  deviations  should  be  greater  than  3.     In  the 
illustrative  example  notice  that  keeping  three  decimal 
places  in  the  average  has  made  more  than  half  of  the 
deviations  greater  than  30.     In  such  a  case  the  average 
could  obviously  be  rounded  off  to  two  decimal  places, 
1.79,  and  at  least  half  the  deviations  would  still  be  greater 
than  3.     This  will  be  found  to  simplify  the  calculation 
and  the  result  will  not  be  essentially  different. 

Find  the  dispersion  of  the  same  ten  variates  by  writing 
a  column  of  deviations  from  the  value  1.79,  and  calculat- 
ing the  value  of  .6745  times  the  square  root  of  one  ninth 
of  the  sum  of  their  squares.  Use  logarithms  but  do  not 
refer  to  the  illustrative  example  in  §  187  for  each  step  of 
the  process;  refer  to  the  formula  (§  185)  instead.* 

Notice  that  your  final  result  agrees  with  the  one 
previously  obtained  as  far  as  three  significant  figures. 
This  is  ample  accuracy,  since  two  significant  figures  are 
all  that  is  usually  wanted  for  the  value  of  a  dispersion. 

190.  Use  of  the  Table  of  Dispersions. — The   root   ex- 
traction and  long  multiplication   and  division  should, 
of  course,  never  be  done  by  the  tedious  arithmetical 
process.     Even  the  logarithmic  process  used  in  the  illus- 

*  Referring  to  the  steps  of  a  typical  example  is  almost  always  the 
easiest  way  of  obtaining  a  required  result;  it  is  a  very  poor  way  of 
earning  a  method  of  procedure. 


XVII  DEVIATION  AND  DISPERSION  217 

tration  can  be  avoided  as  follows:  49010  -s-  9  =  5446; 
the  square  root  of  this  will  be  somewhat  more  than  70; 
f  of  70  is  about  48;  in  the  column  headed  (n  =fc  J)2/.67452 
of  the  table  of  squares  in  the  appendix  find  two  numbers 
between  which  5446  lies  and  read  the  corresponding 
number  in  column  n.  It  is  immediately  seen  to  be  50, 
which  agrees  with  the  previously  obtained  49.78  as  far 
as  the  two  significant  figures  which  are  all  that  is  required. 
As  any  arrangement  of  significant  figures  has  two  square 
roots  (see  §  13,  no.  41,  and  §  86)  a  place  for  5446  would 
also  be  found  opposite  16  in  the  table,  but  the  rough 
preliminary  check-calculation  showed  that  the  answer 
should  be  about  48,  so  there  could  be  no  doubt  that  the 
required  answer  is  50  rather  than  16. 

If  the  sum  of  the  squares  had  been  490100  what  would 
be  the  value  of  the  dispersion,  pointed  off  so  as  to  give 
centimetres  of  length?  Use  the  table  of  squares  in  the 
manner  just  illustrated. 

If  the  sum  of  the  squares  of  the  deviations  of  sixteen 
variates  is  49010  what  is  their  dispersion? 

191.  Dispersions  with  the  Slide  Rule. — In  the  rest  of 
this  course  the  dispersions  are  to  be  calculated  either  with 
the  table  of  squares,  as  explained,  or  with  the  slide  rule, 
which  makes  the  process  even  easier.  A  special  line 
is  marked  at  6745  on  the  C  scale  so  that  .6745  |/a/6 
can  be  obtained  with  a  single  setting  as  soon  as  the  sum 
of  the  squares  of  the  deviations  is  obtained.  Difficulty 
with  the  double  square  root  is  best  avoided  by  setting 
the  end  of  the  C  scale  to  the  approximate  value  of  the 
radical  as  obtained  by  a  rough  mental  calculation;  a 
very  slight  movement  of  the  slide  is  then  all  that  is 
needed  to  make  an  exact  setting. 

If  the  sum  of  the  (v2)'s  of  12  measurements  is  28860 
find  the  approximate  value  of  the  dispersion  mentally, 


218  THEORY  OF  MEASUREMENTS  §193 

and  then  find  the  exact  value  with  one  setting  of  the 
slide  rule. 

192.  Sigma  Notation. — The  capital  letter  sigma  (S)  of 
the  Greek  alphabet  is  often  used  by  mathematicians, 
prefixed  to  an  algebraical  term,  to  denote  the  sum  of  all 
such  terms',  thus  the  expression  for  the  average,  namely, 
(«i  +  a2  +  a3  •  •  •  +  an)/n  is  abbreviated  to  the  equivalent 
form  (2a)/w.  In  the  same  way  the  formula 


.6745 


n  -  1 
is  more  easily  and  compactly  written  in  the  form 


.6745  i/2(v2)!(n  -  1), 


and  the  use  of  2(v2)  will  have  been  noticed  previously 
in  the  example  of  the  calculation  of  a  dispersion. 

Rewrite  the  formulae  for  the  harmonic  mean  and  the 
quadratic  mean  (§  170),  using  the  sigma  notation.  Write 
the  formula  for  (x\  +  x2)z,  using  the  same  notation  as 
far  as  possible. 

If  five  measurements  of  the  quantity  x  give  3,  3,  4, 
5,  10,  what  is  the  value  of  Zz?  If  the  average  is  5 
what  is  the  value  of  Zt>?  Of  2  (mod  »)?*  Of  2(0*)? 
Of  2(z2)? 

193.  Dispersion  of  an  Average.  —  The  average  length 
of  10  variates  has  already  been  calculated.  If  10  more 
were  measured  their  average  would  be  somewhere  nearly 
the  same  as  the  first  average.  If  a  considerable  number 
of  such  averages  had  been  determined  it  would  be  a 
simple  matter  to  determine  their  dispersion,  and  the 
result  would  naturally  be  a  smaller  number  than  the 

*  The  modulus  of  a  (real)  number  means  its  arithmetical  value 
regardless  of  its  sign;  its  "absolute"  value;  the  positive  square 
root  of  its  square. 


XVII  DEVIATION  AND  DISPERSION  219 

dispersion  of  any  of  the  sets  of  individual  measurements, 
since  an  average  is  a  more  trustworthy  figure  than  a 
single  determination.  In  fact  it  can  be  shown  mathe- 
matically (vide  infra)  that  if  ten  equally  good  measure- 
ments are  averaged  the  single  measurements  will  show  a 
variation  which  is  greater  than  the  variation  of  such 
averages  in  the  proportion  of  1/10  to  1,  and  similarly 
that  the  dispersion  of  averages  will  be  I/  \/n  as  great  as 
the  dispersion  of  individual  measurements  if  the  latter  are 
averaged  in  groups  of  n.  This  means  that  it  is  not 
necessary  to  calculate  several  averages  in  order  to  find 
their  dispersion,  for  it  can  be  determined  from  a  knowl- 
edge of  the  number  of  measurements  that  go  to  make 
up  a  single  average  and  from  the  dispersion  of  these 
individual  measurements.  Thus  the  dispersion  of  the 
average  (dav),  as  it  is  called,  of  nine  measurements  is  at 
once  seen  to  be  one  third  as  large  as  the  dispersion  for 
single  measurements  (di),  since  the  square  root  of  nine 
is  three. 

If  a  set  of  16  measurements  are  free  from  constant 
errors  how  much  more  accurate  is  the  average  than  one 
of  the  individual  measurements? 

The  formula  for  the  dispersion  of  an  average  is  easily 
written,  for  if 


then 


the  second  formula  being  I/  V  n  as  large  as  the  first. 

194.  The  Statement  of  a  Measurement.  —  It  is  cus- 
tomary to  write  the  result  of  an  accurate  physical  meas- 
urement in  the  form  of  two  numbers  separated  by  a 
plus-or-minus  sign.  The  first  number  is  the  average; 


220 


THEORY  OF  MEASUREMENTS 


§195 


the  second  one  is  the  dispersion  of  the  average,  not  the 
dispersion  for  single  measurements. 

Tabulate  the  first  eleven  of  the  twelve  measurements 
of  the  wooden  block  made  with  the  card-board  model  of 
a  vernier  caliper.  Pick  out  their  quartiles  and  find  the 
semi-interquartile  range,  to  be  used  as  a  rough  value 
of  the  dispersion  of  the  individual  measurements,  and 
divide  it  by  the  square  root  of  n.  Write  the  thickness 
of  the  block  in  the  form  av.  d=  dav.  Divide  di  for  your 
measurement  of  10  seeds  by  V  10  in  order  to  obtain  dav 
for  them,  and  state  their  measurement  in  the  same  form. 
Make  eleven  more  measurements  of  the  wooden  block, 
this  time  with  the  vernier  caliper  that  gives  tenths  of  a 

millimeter  and  treat  them  in  the 
same  way  as  the  previous  eleven. 
A  typical  set  of  results  is  shown 
in  the  margin.  Two  important 
facts  are  illustrated  by  this  table : 

(1)  It  is  the  accurate    method 
that    shows     discrepancies    be- 
tween   repeated    measurements 
and    the    rough     method    that 
shows  more  uniform  agreement. 

(2)  There  will  b£  nothing  falla- 
cious about  the  statement  that  a 
dispersion    "is  zero"  if    care   is 
taken  to  point  off  that  zero.     The 

semi-interquartile  range  of  the  first  column  is  .0  while 
that  of  the  second  is  .01  cm.,  but  .0  cm.  is  the  only  correct 
way  of  rounding  off  to  one  decimal  place  the  number 
which  is  expressed  in  the  second  decimal  place  by  the 
figures  .01  cm. 

195.  Relative  Dispersion. — It  has  already  been  shown 
that  in  order  to  see  how  serious  an  error  really  is  it 


model  cat. 

steel  cal. 

3.7 

3.75 

3.7 

3.74 

3.7 

3.75 

3.8 

3.76 

3.7 

3.76 

3.7 

3.75 

3.7 

3.76 

3.7 

3.75 

3.7 

3.75 

3.7 

3.74 

3.7 

3.74 

COMPARISON  OF  ROUGH 
MEASUREMENTS  AND  PRE- 
CISE MEASUREMENTS. 


XVII  DEVIATION  AND  DISPERSION  221 

should  be  referred  to  or  divided  by  the  true  magnitude 
of  the  quantity  measured.  In  the  same  way,  instead  of 
using  the  actual  value  of  the  dispersion,  it  is  often  found 
more  useful  to  obtain  the  proportional  dispersion,  or 
relative  dispersion,  or  fractional  dispersion,  as  it  is  also 
called;  the  ratio  of  dispersion  to  representative  magni- 
tude. 

In  the  two  measurements  just  stated,  the  length  of 
a  seed  and  the  thickness  of  the  wooden  block,  divide 
the  dispersion  of  the  average  by  the  average  itself  in 
order  to  obtain  the  relative  dispersion  of  the  average. 
Express  this  either  as  a  decimal  or  as  a  percentage. 
Notice  that  it  is  smaller  for  the  thickness  of  the  block, 
which  is  fairly  uniform,  than  for  the  length  of  a  seed, 
which  varies  considerably.  The  absolute  dispersion  of 
the  average  in  centimetres,  however,  may  be  larger  for 
the  block  than  for  the  seeds  if  a  large  dimension  of  the 
block  is  measured  while  the  seeds  that  are  used  are 
small. 

196.  Questions  and  Exercises. — 1.  If  each  deviation, 
v,  is  a  number  of  thousandths  explain  why  the  mathe- 
matical operations  of  finding  the  dispersion  cause  d  also 
to  be  given  in  thousandths  of  a  unit. 

2.  If  the  results  of  a  series  of  measurements  are  found 
to  be  8.16  cm.  =b  0.033  cm.  would  it  be  advisable  either 
(a)  to  write  8.160  ±  .033,  or  (6)  to  write  8.16  ±  .03, 
instead  of  using  the  first  form?     Explain  why. 

3.  Does  the  number  49010  in  the  table  of  §  187  mean 
490.10  or  4.9010?     Is  it  a  number  of  centimetres  or  of 
square  centimetres? 

4.  Re-arrange  the  following  table  so  that  each  col- 
loquial statement  is  associated  with  its  proper  numerical 
equivalent. 


222  THEORY  OF  MEASUREMENTS  §196 

"a  few  hundred" 20  ±  5 

"nearly  a  gross" 250  ±  50 

"a  dozen  or  so" 70  ±  10 

"upward  of  three  score" 130  ±  10 

5.  Express  each  of  the  following  in  the  form  of  a 
representative  magnitude  and  a  measure  of  scattering: 

"  over  a  dozen  "  "  six  cr  eight " 

" nearly  a  hundred "  "a  few " 

' '  about  a  thousand  "  "  several ' ' 

"  iii  the  neighborhood  of  15  or  20  "  "  some  " 

6.  If  "  average  deviation  "  means  the  average"  of  the 
(positive)  arithmetical  values   of  the  deviations,  what 
would  probably  be  meant  by  the  expression  "median 
deviation"?     Has  any  characteristic  deviation  already 
been  named  which  has  practically  the  same  significance? 


XVIII.     THE    WEIGHTING   OF   OBSERVATIONS 

Apparatus. — Platform  balance;  clamp  and  bar  or  stand 
to  support  the  balance  40  or  50  cm.  above  the  table; 
set  of  weights;  vernier  caliper;  aluminum  block;  over- 
flow can  and  catch-bucket  for  measuring  displaced  water; 
towel;  string  (80  to  100  cm.)  and  two  spreading  rods; 
fine  silk  thread;  slide  rule. 

197.  Necessity  of  Weights  for  Observations. — A   rep- 
resentative   value   is    often   wanted   for    measurements 
which  are  not  all  equally  trustworthy.     The  accepted 
values  for  such  constants  as  the  maximum  density  of 
water,  the  mechanical  equivalent  of  heat,  the  length  of 
the  true  ohm  of  mercury,  the  velocity  of  light  in  vacuo, 
have  all  been  derived  from  measurements  by  different 
observers  at  various  times,  and  in  general  by  different 
apparatus  and  methods.     Any  of  these  varying  factors 
will   produce   varying   results,    and   one   determination 
can  sometimes  be  accepted  with  more  confidence  than 
another,  and  so  will  be  entitled  to  greater  " weight" 
when  it  is  necessary  to  decide  upon  a  representative  value. 

198.  Density  by  Different  Methods. — An  example  of 
the  effect  of  different  methods  on  the  determination  of  a 
physical  magnitude  may  be  given  by  the  measurement 
of  the  density  of  a  metal  block.     If  the  mass  is  known 
this  can  be  accomplished  either  by  mensuration,  or  by 
measuring  displacement,  or  by  a  measurement  of  buoyant 
force.     According  to  the  Principle  of  Archimedes  the 
apparent  loss  of  weight  of  a  body  immersed  in  a  fluid  is 
the  same  as  the  weight  of  an  equal  volume  of  the  fluid. 
If  the  volume  of  a  metal  block  is  v,  its  weight  w,  and  its 
apparent  weight  in  water  w',  the  density  can  be  found 

223 


224  THEORY  OF  MEASUREMENTS  §198 

as  the  ratio  of  the  weight,  w,  to  the  loss  of  weight, 
w  —  wf,  supposing  that  the  density  of  water  is  unity; 
or  it  can  be  determined  as  the  ratio  of  the  weight,  w, 
to  the  measured  volume  of  water  that  is  actually  dis- 
placed on  immersion,  say  v',  or  the  block  can  be  meas- 
ured with  a  caliper  and  the  density  calculated  as  m/v. 

It  will  first  be  necessary  to  arrange  the  apparatus  so 
that  the  apparent  weight  of  an  object  can  be  determined 
while  it  is  immersed  in  water.  Place  the  platform 
balance  on  the  support  or  clamp  it  to  the  cross-bar 
above  the  table  in  such  a  way  that  an  object  can  be 
weighed  by  suspending  it  under  the  bar  with  strings 
attached  to  a  spreading  rod  that  is  laid  on  one  scale-pan 
of  the  balance.  See  that  the  beam  of  the  balance  moves 
freely.  Use  the  other  spreading  rod  as  a  counterpoise, 
and  make  a  careful  allowance  for  the  fact  that  they  may 
not  exactly  balance.*  Attach  the  aluminum  block  to 
the  string  by  a  fine  thread  long  enough  to  allow  it  to 
hang  within  the  empty  overflow  can,  and  (1)  weigh  it  as 
accurately  as  possible.  Fill  the  overflow  can  with  water, 
closing  the  spout  with  the  finger-tip;  place  it  in  position 
where  the  aluminum  block  is  to  hang,  with  the  catch- 
bucket  under  the  spout;  remove  the  finger  and  allow 
the  excess  of  water  to  run  out  of  the  overflow  can; 
then  (2)  weigh  the  catch-bucket  with  its  contained  water, 
and  replace  it  in  position.  Lower  the  aluminum  block 
carefully  into  the  overflow  can  and  (3)  weigh  it  while 
submerged;  then  (4)  weigh  the  catch-bucket  again  in 
order  to  find  out  how  much  water  was  displaced. 

Find  the  density  of  the  aluminum  block  (a)  by  com- 
paring its  weight  with  the  weight  of  the  overflow  of 
water  actually  displaced;  (6)  from  the  two  values  w  and 
w';  (c)  by  measuring  the  block  with  the  vernier  caliper, 

*  Weigh  their  difference;  there  is  no  need  of  weighing  each  one 
separately. 


XVIII        THE  WEIGHTING  OF  OBSERVATIONS  225 

computing  its  volume  as  closely  as  possible,  and  applying 
the  formula  for  density,  d  =  m/v.  Report  your  results 
to  the  instructor  for  comparison  with  those  of  the  other 
members  of  the  class. 

199.  Weights  for  Repeated   Values. — The   simplest 
case  of  weighting  different  observations  is  when  separate 
numerical  values  have  each  been  obtained  a  definite 
number  of  times.     Suppose,  for  example,  that  the  density 
of  a  block  of  aluminum  has  been  determined  both  as  2.6 
and  as  2.7,  in  the  total  of  five  measurements,  the  smaller 
value  having  been  found  on  four  occasions  while  the 
larger  value  was  obtained  only  once.     The  best  repre- 
sentative figure  from  these  data  certainly  would  not  be 
the  number  2.65,  half  way  between  2.6  and  2.7,  but 
ought  to  be  a  number  situated  four  times  as  far  from  the 
least    frequent    measurement,    2.7,    as    from   the    most 
frequent  one,   2.6;    in  other  words,   it  should  be  the 
number  2.62.     Moreover,  this  is  easily  seen  to  be  the 
same  result  as  would  be  obtained  by  taking  the  average 
of  the  five  individual  measurements.     (Try  it.)     The 
rule  in  such  a  case  is  obviously  to  give  each  numerical 
value  a  weight  proportional  to  the  number  of  times  of  its 
occurrence. 

Find  the  weighted  average  of  the  values  of  a  measured 
length  if  it  was  found  to  be  2.345  cm.  in  each  of  six 
trials',  2.350  cm.  in  twelve  trials,  and  2.355  in  nine  trials. 
(Suggestion:  it  is  a  little  easier  to  calculate  the  .value  of 
2  X  2.345  +  3  X  2.355  +  4  X  2.350.) 

200.  The  Weighted  Average. — The  weighted  average 
is  found  in  any  case  by  considering  that  certain  values 
have  been  obtained  more  frequently  than  others.     In 
the  case  just  discussed  this  was  a  fact,  in  other  cases  it 
is  only  a  supposition  made  to  fit  the  known  or  estimated 
intrinsic  value  of  the  observations. 

16 


226  THEORY  OF  MEASUREMENTS  §120 

If  a  difficult  measurement  had  been  made  by  an  ex- 
perienced student  and  found  to  be  0.35,  while  the  same 
experiment  gave  the  value  0.41  when  performed  by  a 
beginner,  it  might  be  decided  somewhat  arbitrarily  to 
give  the  first  number  twice  the  weight  of  the  second. 
The  process  of  finding  the  weighted  average,  (2  X 
0.35  +  1  X  0.41)/3,  would  then  be  equivalent  to  sup- 
posing that  the  better  measurement  had  bden  obtained 
on  two  occasions  but  the  poorer  one  only  once.  If 
a  measurement  of  some  quantity  had  been  found  to  be 
1.36  when  made  under  unfavorable  circumstances,  and 
1.41  when  made  under  circumstances  that  were  more 
favorable  to  experimentation  it  might  be  considered 
best  to  assign  the  respective  weights  of  1  and  1.5  to 
the  two  values.  The  weighted  average  would  then  be 
(2  X  1.36  +  3  X  1.41)  4-  5,  or  1.39,  a  figure  which  will 
be  seen  to  be  nearer  to  the  better  value  than  to  the  poorer 
one  in  exactly  the  ratio  of  1  to  1.5. 

201.  Arbitrarily  Assigned  Weights. — The  objection- 
able feature  of  such  an  arbitrary  assignment  of  weights 
is  very  obvious.  The  relative  weights  depend  too  much 
upon  the  judgment  of  the  individual  computer;  further- 
more, it  is  often  difficult  to  avoid  being  influenced  by 
the  fact  that  certain  determinations  vary  more  or  less 
widely  from  the  expected  value,  instead  of  keeping  one's 
judgment  focussed  on  the  quality  of  the  experimental 
work. 

Which  do  you  consider  the  better  method  of  determin- 
ing density,  by  buoyancy,  or  by  displacement?  Choose 
what  you  consider  the  best  ratio  for  their  relative  accu- 
racies and  find  the  corresponding  weighted  average,  but 
be  careful  not  to  give  extra  weight  to  either  measure- 
ment on  account  of  its  coming  close  to  the  third  deter- 
mination made  by  calculating  the  volume  obtained  by 
mensuration  (see  §  159,  If  3). 


XVIII        THE  WEIGHTING  OF  OBSERVATIONS  227 

202.  Weight  and  Dispersion. — Determinations  of  any 
carefully  measured  magnitude  are  usually  stated  in  the 
form  of  an  average  and  its  dispersion,  a  d=  dav.  Subject 
to  the  condition  that  the  influence  of  constant  errors 
can  be  neglected,  it  can  be  shown  mathematically  that 
the  best  value  for  a  measurement  is  obtained  by  weighting 
each  determination  of  an  average  in  inverse  proportion  to 
the  square  of  its  dispersion.  Thus,  if  one  determination 
has  a  dispersion  of  .0040  and  another  has  a  dispersion 
of  .012  the  former  should  be  given  nine  times  as  much 
weight  as  the  latter.  This  can  be  expressed  in  a  general 
formula,  if  d  is  used  to  denote  the  dispersion  of  an 
average,  by  saying  that  the  weighted  average  of 

cti  d=  di,  a%  it  dz,  a^  d=  d^, 
is 


or 

w.  av.  =  Z 

but  it  is  much  better  to  learn  the  principle  involved 
than  to  memorize  the  formula. 

203.  Limitations  of  w  =  k/d2. — Attention  should  again 
be  directed  to  the  fact  that  weighting  according  to  dis- 
persions takes  no  account  of  the  fact  that  constant  errors 
may  be  present  in  the  given  data.  The  dispersion  sum- 
marizes only  the  accidental  errors,  and  if  the  constant 
errors  are  greater  than  these  the  weighted  average  is 
no  better  than  the  simple  arithmetical  average  (Figs.  55, 
56). 

Tabulate  the  determinations,  made  by  the  various 
members  of  the  class,  of  the  density  of  aluminum  as 
found  by  the  effect  of  buoyancy.  Calculate  the  typical 
value  in  the  form  a  d=  d. 


228  THEORY  OF  MEASUREMENTS  §204 

Find  in  the  same  way  the  average  and  dispersion  of 
the  density  as  determined  by  displacement.  Use  the 
slide  rule,  not  the  tables  of  logarithms. 

Calculate  the  weighted  average  of  these  two  data. 


55  55 

/ 

FIGS.  55,  56.  TARGET  DIAGRAMS  ILLUSTRATING  ERRORS. — In 
Fig.  55  the  two  groups  have  nearly  the  same  center,  that  with  the 
smaller  dispersion  naturally  being  the  more  trustworthy.  In  Fig. 
56  at  least  one  of  the  groups  shows  such  a  large  constant  error  that 
the  relative  difference  in  the  two  sets  of  accidental  errors  is  un- 
important. 

There  are  several  sources  of  constant  error  in  each 
of  the  two  above  methods  of  determining  density.  State 
at  least  four  that  are  common  to  both  methods,  and  at 
least  one  that  influences  one  form  of  experiment  but 
not  the  other.  For  example,  if  bubbles  cling  to  the 
block  when  it  is  immersed  (it  is  usually  difficult  to  avoid 
them  altogether)  the  apparent  weight  will  be  lessened, 
always  causing  the  calculated  density  to  be  too  low; 
they  will  also  cause  too  much  water  to  overflow,  again 
making  the  calculated  density  lower  than  it  should  be. 
Consider  also  the  effects  of  such  things  as  inequality  of 
the  beam-arms  of  the  balance,  capillary  attraction  where 
the  thread  cuts  the  surface  of  the  water,  etc. 

204.  Exception  to  the  Rule. — The  method  of  weight- 
ing observations  in  inverse  proportion  to  their  dispersions 
is  used  for  separate  and  independent  data  whose  relative 
accuracy  is  assumed  to  be  shown  by  their  dispersions. 
Where  two  or  more  series  of  observations,  however, 
are  known  to  have  been  made  with  equally  trustworthy 
apparatus,  methods,  and  observers  they  should  be 


XVIII        THE  WEIGHTING  OF  OBSERVATIONS  '          229 

weighted  merely  according  to  the  number  of  measure- 
ments which  each  comprises,  notwithstanding  that  their 
dispersions  might  indicate  a  very  different  result.*  To 
do  otherwise  would  be  to  repudiate  the  principle  of  the 
average,  which  depends  upon  the  fact  that  all  observa- 
tions are  supposed  to  be  equally  trustworthy.  On  the 
other  hand,  when  different  observations  are  known  to 
be  unequally  trustworthy,  even  if  they  occur  in  the 
same  series,  weight  may  be  given  to  the  fact  that  some 
are  closely  clustered  about  an  apparent  central  position 
while  others  diverge  erratically.  A  great  advantage  of 
the  median,  as  a  representative  magnitude,  is  that  it  is 
not  unduly  influenced  by  a  very  large  or  a  very  small 
measurement  and  hence  it  automatically  gives  less  weight 
to  the  more  aberrant  measurements  of  a  given  group. 

Which  is  to  be  preferred,  the  average  or  the  median, 
for  a  determination,  like  the  one  just  made,  of  density 
by  buoyancy?  Why? 

205.  Questions  and  Exercises. — 1.  Why  was  the 
aluminum  block  hung  with  a  thread  instead  of  a  string? 
Would  there  be  any  advantage  in  using  wire  instead? 
If  wire  should  be  used  what  kind  of  wire  would  be  best? 

2.  Is  it  necessary  to  calculate  dispersions  in  order  to 
weight  averages  in  accordance  with  §  202.     What  simpler 
calculation  can  be  used  to  give  exactly  the  same  result? 

3.  Read  §  186  again,  and  write  in  your  note  book  a 
statement  of  the  most  important  fact  that  it  contains. 
Do  not  follow  the  form  of  any  of  the  printed  statements, 
but  try  to  write  out  the  fact  from  an  essentially  different 
point  of  view. 

*  Similarly,  a  single  set  of  measurements  known  to  be  uniformly 
and  equally  good  must  be  simply  averaged;  to  give  more  weight  to 
the  individual  measurements  that  have  the  smaller  deviations  would 
be  a  procedure  akin  to  substituting  the  median  for  the  average  (but  , 
see  also  the  end  of  §  174). 


XIX.     CRITERIA   OF   REJECTION 

Apparatus. — Slide  rule. 

206.  Observational  Integrity. — When  successive  re- 
determinations  of  a  quantity  have  beeri  made  in  the 
course  of  an  experimental  investigation  it  is  to  be  sup- 
posed that  they  have  all  been  made  with  an  equal 
degree  of  care. 

It  is  important  to  remember  that  an  observation 
should  never  be  rejected  simply  because  it  is  not  in 
satisfactory  agreement  with  the  other  determinations 
of  the  series.  If  the  experimenter  realizes  that  one  of 
his  measurements  was  made  under  some  kind  of  a  handi- 
cap or  under  such  conditions  that  a  faulty  result  would 
be  likely  it  is  permissible  to  cross  out  the  corresponding 
value  in  his  notes  and  to  omit  it  in  the  final  consideration 
of  the  cjata,  but  there  must  be  some  definite  and  satis- 
factory reason  for  discarding  it  other  than  the  fact  of 
its  divergence  from  the  expected  value.  The  tempta- 
tion, often  felt  by  the  beginner,  to  omit  or  "  re-deter- 
mine"* a  discordant  result  may  be  very  perceptible, 
but  absolute  freedom  from  prejudice  (see  dependent 
measurements,  §  159;  also  §  47,  j[  3)  should  be  cultivated 
to  such  a  point  that  the  experimenter  is  habitually 
able  to  feel  a  certain  disinterestedness  in  the  outcome 
of  a  measurement  after  he  has  first  taken  pains  to  ensure 
its  being  as  trustworthy  as  possible.  His  attitude  should 
be,  "I  have  done  all  that  can  be  done  in  the  way  of 
preparations  for  making  this  measurement  accurate  and 
independent;  now  let  the  result  turn  out  as  it  will." 

*  A  re-determination  is  not  intrinsically  objectionable,  but  it 
should  be  made  in  addition  to  the  other  determination,  not  in  place 
of  it. 

230 


XIX  CRITERIA   OF  REJECTION  231 

207.  Importance  of  Criteria. — Even  with  all   care  to 
make  successive  measurements  equally  accurate  it  often 
happens  that  one  or  more  of  them  show  unduly  large 
deviations  from  the  average.     In  order  to  prevent  these 
values  from  having  an  abnormal  influence  on  the  repre- 
sentative value  certain  rules  have  been  formulated  for 
determining  whether  they  shall  be  retained  or  discarded, 
for  if  an  observer  merely  used  his  own  judgment  in 
deciding  the  question  the  result  would  depend  too  much 
upon  his  own  individuality  and  temperament,  and  dif- 
ferent observers  would  obtain  different  results  from  data 
identically  the  same,  just  as  in  the  case  of  the  arbitrary 
assignment  of  weights  (§  201).     In  fact,  the  rejection  of 
a  measurement  is  nothing  more  nor  less  than  giving.it 
a  weight  equal  to  zero. 

208.  Chauvenet's   Criterion  — One   of  the   easiest   to 
understand  of  the  various  devices  for  testing  doubtful 
observations  is  known  as  Chauvenet's  criterion  of  rejection, 
according   to    which   rejectability   is    determined    as    a 
function  of  deviation,  dispersion,  and  number  of  meas- 
urements.    An  unduly  large  deviation  is  an  argument 
for  rejection,   especially  if  the  dispersion  is  relatively 
small;    furthermore,  a  deviation  so  large  that  it  would 
not  be  expected  to  occur  more  than  once  out  of  a  hundred 
cases  might  not  seriously  affect  the  average  of  a  hundred 
values,  but  if  it  happened  to  occur  in  a  set  of  only  ten 
measurements  it  would  probably  exercise  an  altogether 
disproportionate  effect  upon  their  average.     The  object 
of  a  criterion  of  rejection  is  not  to  indicate  that  a  certain 
measurement  is  wrong,  but  merely  to  point  out  that  it 
is  liable  to  be  misleading  it  if  occurs  among  a  small 
group.* 

*  Remarkably  wide  deviations  may  be  expected  to  occur  once 
in   a  while  if  the  number  of  measurements  is  extremely  large; 


232  THEORY  OF  MEASUREMENTS  §208 

The  following  exercise  is  for  the  purpose  of  showing 
the  theory  of  the  effect  that  the  combined  influences  of 
deviation,  dispersion,  and  number  of  meas- 
urements exert  upon  the  determination  of 
the  advisability  of  keeping  or  rejecting  any 


50.0 


49.0 


49.6  Draw  a  graphic  diagram  from  the  table. 

Then  draw  the  ordinates  x  =  10  and  x  =  50 
47.2       from  the  base  line  up  to  the  curve.     The 


46.0 


l      49  9       single  measurement  of  a  group." 

2 

3 

4 

5 

6 

7 

8 

9 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42, 
44 
46 
48 
50 
52 
54 
56 
58 
60 


44;6       result  will  be  the  right-hand  half  of  a  nor- 
43.3       mal  frequency  diagram,  the  ^-values  corre- 
39^9       sponding  to  deviations  and  the  ^/-values  to 
36.0       the  frequency  of  their  occurrence.     Remem- 
2s!o       ber  that  the   total    area  of   such  a  curve 
24.0       corresponds  to   the  number  of  deviations 
(§  164);  in  the  same  way.  the  area  between 
13.8       curve  and  base  line  which  is  bounded  on 
the  left  and  right  by  any  two  ordinates  rep- 
resents  the  number  of  observations  whose 


108 


* 


3*6       numerical  deviations  lie  between  those  two 
limits.     As  the  dispersion  is  the  same  as  the 
median  deviation  (§  186)  it  is  evident  that 
°-9       the  ordinate  which  bisects  the  area  (this  is 
0:5       the  ordinate  x  =  10,  for  the  scales  used  in 
this  .diagram)  must    have  its  abscissa  nu- 
O.i       merically  equal  to  the  dispersion. 

Suppose  another  ordinate  is  drawn  at 
o!o  a  position  so  far  to  the  right  that  it 
°-°  includes  between  the  ?/-axis  and  itself 


VALUES  OF      nine    tenths  of  the  total  area   under  the 

2/=50e-(.o4769*)2    curve   and  leaves  only   one  tenth  of   the 

area  beyond  it  to  the  right,  then  the  cor- 

notice  that  there  is  some  space  between  the  curve  y  =  e~*z  and  the 
x-axis  even  at  a  great  distance  from  the  i/-axis  (§  §67,107). 


XIX  CRITERIA   OF  REJECTION  233 

responding  abscissa  would  similarly  have  a  value  that 
would  be  exceeded  by  only  one  tenth  of  the  total  num- 
ber of  deviations,  and  if  any  one  deviation  were  chosen 
at  random  there  would  be  only  one  chance  in  ten  that 
it  would  be  larger  than  the  corresponding  z-value.  It 
can  be  proved  mathematically*  that  in  order  to  include 
nine  tenths  of  the  area  the  ordinate  must  be  drawn  2.44 
times  as  far  to  the  right  of  the  y-axis  as  the  line  which 
bisects  the  area  and  corresponds  to  the  dispersion. 

On  your  diagram  draw  the  ordinate  that  includes 
nine  tenths  of  the  area  and  make  sure  that  its  abscissa 
fulfills  the  condition  stated  above.  If  the  total  number 
of  measurements  were  ten  how  many  would  most  prob- 
ably be  represented  by  the  area  to  the  right  of  ,the 
ordinate?  How  many  if  the  number  of  measurements 
were  50?  How  many  if  the  number  were  4?  How 
many  if  6?  The  last  two  questions  should  be  answered 
to  the  nearest  whole  number. 

Since  the  ordinate  for  x  =  2.44  d  includes  nine  tenths 
of  the  area  and  the  limit  2.44  X  dispersion  includes 
nine  tenths  of  the  deviations  it  might  be  said  theoret- 
ically and  rather  figuratively  that  if  there  were  only 
five  measurements  in  a  certain  series  the  number 
of  measurements  whose  deviations  were  greater  than 
this  limit  would  most  probably  be  just  one  half.  In 
other  words  the  limit  would  be  just  on  such  a  border 
line  that  if  it  were  decreased  we  should  expect  it  to 
exclude  one  measurement  rather  than  no  measurements, 
and  if  it  were  increased  we  should  expect  it  to  exclude 
no  measurements  rather  than  one  measurement. 

*  Certain  statements  are  intended  to  be  taken  on  faith  by  the 
student.  This  one  is  just  as  true  as  the  statement  that  the  fre- 
quencies of  accidental  errors  follow  the  law  y  =  e~*2  or  that  fluid 
friction  in  a  water  pipe  varies  as  the  1.8  power  of  the  radius;  all  of 
them  can  be  proved  but  none  of  the  proofs  are  necessary  here. 


234 


THEORY  OF  MEASUREMENTS 


£08 


It  follows,  then,  that  no  one  of  a  series  of  five  measure- 
ments theoretically  ought  to  have  a  deviation  of  more 
than  2.44  times  the  dispersion.  This  being  the  case  it 
is  only  natural  to  consider  that  one  is  justified  in  dis- 
carding any  one  of  the  measurements  of  a  series  of  five 
if  its  deviation  does  exceed  this  limit.  Just  as  the 
ordinate  for  2.44  d  excludes  one  tenth  of  the  area  (i.  e., 
excludes  half  a  measurement  if  there  are  five  in  all)  so 
the  ordinate  at  2.57  d  excludes  one  twelfth  of  the  area, 
which  would  correspond  to  half  a  measurement  if  the 
total  number  of  measurements  were  six.  Accordingly 
no  measurement  in  a  set  of  six  should  theoretically  have 
v  >  2.57  d  if  the  set  is  supposed  to  follow  the  law  of 
frequency  distribution  for  accidental  errors.  Chauvenefs 
criterion  is  simply  an  extension  of  this  delimitation 
to  other  values  of  n  as  well  as  5  and  6.  The  column,  /, 
of  the  following  table  shows  the  limiting  z-value  for 


n 

l 

log* 

„  |  , 

log  i 

n 

i 

log/ 

n 

i 

log/ 

I 

2 
3 
4 
5 

1.00 
1.71 
2.05 
2.27 
2.44 

000 
233 
312 
356 

387 

11 
12 
13 
14 
15 

2.97 
3.02 
3.07 
3.11 
3.15 

473 
480 
487 
493 
498 

21 
22 
23 
24 
25 

3.35 
3.38 
3.40 
3.43 
3.45 

525 
529 
532 
535 
538 

32 

34 
36 

38 
40 

3.59 
3.62 
3.65 
3.68 
3.70 

554 
556 
561 
566 
570 

6 

7 
8 
9 
10 

2.57 
2.67 
2.76 

2.84 
2.91 

410 
427 
441 
453 
464 

16 
17 

18 
19 
20 

3.19 
3.22 
3.26 
3.29 
3.32 

504 
508 
513 
517 
521 

26 

27 
28 
29 
30 

3.47 
3.49 
3.51 
3.53 
3.55 

540 
543 
546 
549 
551 

49 
64 
81 
100 
671 

3.81 
3.95 
4.06 
4.16 
5.00 

581 
597 
608 
619 
699 

Chauvenet's  Criterion. — If  the  most  divergent  measurement  out 
of  a  series  of  n  determinations  has  a  deviation  more  than  I  times 
as  great  as  the  dispersion  of  the  individual  measurements  it  should 
be  rejected. 


which  the  corresponding  part  of  the  area  (see  appendix) 
included  under  the  curve  y  =  e~x*  is  1  —  l/2n  and  the 


XIX  .      CRITERIA   OF  REJECTION  235 

excluded  part  is  half  of  1/w,  as  before,  this  value  being 
expressed  in  terms  of  the  dispersion. 

It  should  be  carefully  kept  in  mind  when  considering 
any  criterion  of  rejection  that  we  are  interested  in  the 
individual  measurements,  and,  accordingly,  the  disper- 
sion to  which  the  criterion  applies  is  the  dispersion  of  the 
individual  measurements,  not  the  dispersion  of  the  aver-, 
age.  Chauvenet's  criterion,  then,  is  the  test  of  whether 
any  deviation  is  greater  than  I  times  the  dispersion  of  the 
individual  values  of  a  series  of  n  measurements,  where  I 
corresponds  to  n  in  the  way  shown  in  the  table. 

209.  The  Probable  Error. — By  this  time  the  student 
ought  to  be  thoroughly  aware  of  the  fact  that  the  dis- 
persion is  not  properly  an  error,  but  a  deviation.     If  he 
also  realizes  that  deviations  within  its  limits  are  no  more 
probable  than  improbable  there  can  be  no  objection  to 
his  using  the  term  that  is  always  employed  by  physicists 
in  speaking  of  this  characteristic  deviation.     In  this  book 
the  term  dispersion  has  been  used  in  order  to  avoid 
repeatedly  informing  the  student  that  it  is  an  error  and 
repeatedly  suggesting  that  there  is  something  very  prob- 
able about  it.     It  will  hereafter  be  spoken  of  as  the 
probable  error,  and  of  course  it  will  be  under- 
stood that  it  is  used  in  two  forms,  the  prob-        'o309 
able  error  of  the  individual  measurements  (di,        -031 
or  pi)  and  the  probable  error  of  the  average        '9347 
W»,  or  p..).  :o|5 

In  an   experimental   determination   of  the         ]038 
specific  heat  of  lead  shot  the  following  values        -045 
were  obtained  by  a  class  of  students.     Test 
them  by  Chauvenet's  criterion  to   determine       SPECIFIC 

whether  any  measurement  falls  outside  of  the  T  E  A  ^ 

.     ,    i .     .  .  -  ,     LEAD  SHOT. 

theoretical  limits,  but  if  two  or  more  such 

values  are  found  reject  only  the  most  divergent  one,  find  a 


236  THEORY  OF  MEASUREMENTS  §210 

new  average  for  those  that  remain,  and  apply  the  cri- 
terion to  them  in  turn.*  Repeat  the  process,  if  necessary, 
until  no  more  values  can  be  discarded,  and  then  state 
the  best  value  obtainable  from  the  figures,  with  its 
"  probable  error." 

210.  Graphic  Approximation  to  Chauvenet's  Criterion. 
— Where  a  graphic  diagram  is  to  be  used  for  only  a 
single  series  of  numbers  instead  of  for  sets  of  values  of 
two  varying  quantities  it  is  advisable  to  use  a  horizontal 
scale  and  lay  off  the  individual  measurements  as  small 
dots  or  circles  (Fig.  57)  unless  they  are  sufficiently 
numerous  to  allow  a  good  histogram  to  be  drawn. 

Q1  ME    Q3 


• 

1 

50 

54 

58        62 

66 

FIG.  57.  DISTRIBUTION  OF  A  FEW  MEASUREMENTS. — When  the 
majority  of  frequencies  are  very  small  each  measurement  may  be 
represented  by  a  dot  instead  of  the  formal  square  of  the  histogram. 
The  centre  of  gravity  of  such  a  system  of  equal-sized  dots  is  more 
readily  apparent  to  the  eye  than  is  the  position  of  the  line  that 
would  be  needed  to  bisect  the  area  of  the  frequency  polygon  or 
histogram. 

The  following  figures  are  the  experimental  values  of 
the  slope  of  the  first  " black-thread"  diagram  as  obtained 
by  a  class  of  students:  .60,  ,57,  .64,  .53,  .48,  .59,  .59,  .61. 
Make  a  graphic  diagram  of  these  values  and  mark  the 
median  and  quartiles.  Do  not  measure  the  semi-inter- 
quartile range  nor  multiply  it  by  I,  but  mark  off  its 
length  on  the  edge  of  a  strip  of  paper  and  apply  this  to 

*  If  one  large  measurement  and  one  small  one  are  beyond  the 
limits  the  rejection  of  the  more  divergent  one  may  result  in  shifting 
the  average  far  enough  toward  the  other  one  to  bring  it  within 
bounds.  Where  it  is  obvious  that  this  cannot  happen  it  is  un- 
necessary to  adhere  to  the  letter  of  the  rule. 


XIX 


CRITERIA  OF  REJECTION 


237 


your  diagram,  laying  it  off  to  the  right  and  to  the  left 
of  the  median*  as  many  times  as  may  be  indicated,  by 
the  criterion.  In  this  way  a  rough  application  of 
the  criterion  can  be  made  graphically  and 
the  long  calculation  can  be  avoided.  De- 
termine from  the  diagram  whether  any 
value  should  be  rejected  and  then  verify 
the  result  by  the  usual  form  of  calculation. 

Use  the  graphic  method  for  applying 
Chauvenet's  criterion  to  the  following  set 
of  barometer  readings : 

What  advantage  has  the  arithmetical 
method  over  the  graphic  method? 

Write  down,  in  your  own  words,  just  what 
it  is  that  is  represented  by  (a)  the  probable 
error  of  a  single  measurement,  .6745 


29.986  inches 

29.982 

29.990 

29.984 

29.984 

29.980 

29.986 

29.977 

29.984 

29.982 

29.986 

29.988 

29.984 

BAROMETER 
READINGS. 

(n  -  1),  and  by 
(6)  the  probable  error  of  the  average,  .6745  ^2d?/n(n  —  1). 

211.  Irregularities  of   Small  Groups. — The  probable 
error,  or  "dispersion,"  cannot  be  considered  as  having 
much  meaning  in  cases  where  the  total  number  of  meas- 
urements is  less  than  ten,  and  even  with  ten  measure- 
ments it  should  be  treated  with  a  certain  amount  of 
caution.     A  number  of  values  less  than  ten  will  hardly 
ever  give   a  histogram  of  their  frequency-distribution 
which  is  recognizably  similar  to  the  graph  of  y  =  e~x*, 
the   curve   which   all   unbiassed   measurements  will   be 
found  to  follow  if  they  are  sufficiently  numerous. 

212.  Justification   of   the    Criterion. — For    the    same 
reason,  it  is  hardly  worth  while  to  use  a  criterion  of 
rejection  for  less  than  ten  measurements.     The  example 
given  above  with  only  five  is  intended  merely  for  an 
illustration  of  the  method  of  using  the  criterion,  and  the 

*  It  is  better  to  use  the  mid-quartile  point  than  the  median  in 
case  the  two  are  not  the  same. 


238  THEORY  OF  MEASUREMENTS  §214 

still  smaller  values  in  the  table  are  only  of  theoretical 
importance.  Chauvenet's  criterion  is  not  to  be  con- 
sidered as  showing  that  any  one  measurement  is  a 
mistake,  but  only  as  indicating  that  a  very  large  deviation 
is  such  a  rarity  that  it  Would  have  an  unduly  large 
influence  upon  the  average  if  it  were  allowed  to  remain 
along  with  the  other  values  of  a  very  limited  series  of 
measurements  (§  208,  page  231,  foot-note). 

213.  Wright's  Criterion. — Another  criterion  of  rejec- 
tion, which  is  sometimes  employed,  is  that  of  Wright. 
According  to  this  the  arbitrary  rejection  of  a  single  meas- 
urement may  be  considered  if  its  deviation  is  more  than 
five  times  the  probable  error. 

Turn  back  to  the  graphic  diagram  of  the  table  in  §  208, 
and  notice  how  small  a  part  of  the  area  of  the  curve  lies 
to  the  right  of  the  ordinate,  x  =  50,  which  represents 
five  times  the  probable  error.  Turn  to  the  table  of 
values  for  Chauvenet's  criterion  and  note  how  many 
measurements  would  need  to  be  made  before  "half  a 
measurement"  would  be  likely  to  diverge  from  the  aver- 
age five  times  as  far  as  the  probable  error. 

In  the  measurements  to  which  you  have  already  applied 
Chauvenet's  criterion  how  many  would  have  been  rejected 
if  Wright's  criterion  had  been  used  instead?  If  a  devi- 
ation is  great  enough  to  be  practically  sure  of  rejection 
by  one  of  the  two  criteria  will  it  ordinarily  be  rejected 
by  the  other?  If  a  maximum  deviation  is  small  enough 
to  avoid  rejection  by  one  criterion  will  it  be  practically 
certain  to  escape  rejection  by  the  other?  Explain  why. 
What  are  the  relative  advantages  of  the  two  criteria? 

214.  Comparison  of  Characteristic  Deviations.— Other 
limiting  values,  which  give  practically  the  same  result 
as  Wright's  criterion,  are  four  times  the  average  deviation, 
and  three  times  the  standard  deviation. 


XIX  CRITERIA  OF  REJECTION  239 

Turn  back  to  your  notes  on  the  use  of  logarithms  and 
find  the  graphic  diagram  of  y  =  e~x*.  Mark  off  the 
following  values  on  the  base  line,  p  =  .4769363,  a  = 
.5641895  and  s  =  .7071066.  These  represent  respec- 
tively the  probable  error,  the  average  deviation,  and  the 
standard  deviation,  and  are  roughly  proportional  to 
10  :  12  :  15;  a  better  approximation  to  their  ratios  than 
is  given  by  10  :  12  :  15  may  be  found  with  the  aid  of 
the  slide  rule.  Draw  the  corresponding  ordinates,  and 
notice  that  the  last  one  meets  the  curve  at  the  point  of 
inflection,  that  is,  at  the  point  where  it  is  momentarily 
straight  as  it  changes  from  convex  upward  to  convex 
downward. 

215.  Questions  and  Exercises. — 1.  Write  a  definition 
of  Chauvenet's  criterion  in  your  own  words.  Notice 
that  the  last  sentence  of  §  208  is  not  a  satisfactory  defini- 
tion. 

2.  Can  a  set  of  three  measurements  comprise  such 
values  that  one  of  them  will  be  rejected  by  Chauvenet's 
criterion?     Give  an  example  to  illustrate  your  answer. 

3.  Can   a   set   of  two   measurements   comprise   such' 
values  that  one  of  them  will  be  rejected  by  Chauvenet's 
criterion.     Give  an  example  to  illustrate  your  answer. 


XX.     LEAST  SQUARES 

Apparatus. — Slide  rule;  black  thread. 

216.  The  Average  as  a  Least-Square  Magnitude. — The 
mathematical   principle   of  least   squares  is   that   when 
measurements  are  equally  trustworthy  their  best  repre- 
sentative value  is  that  for  which  the  sum  of  the  squares 
of  the  deviations  has  the  lowest  numerical  value.*     It 
is  upon  this  principle  that  the  use  of  the  average  is  based, 
for  it  is  easy  to  show  that  the  sum  of  the  squares  of  the 
deviations   of   any   particular  set   of   numbers   will   be 
greater  when  measured  from  some  other  value  than  when 
measured  from  the  average. 

Find  the  average  of  the  numbers  3,  3,  4,  5,  10;  also 
their  deviations  from  the  average,  and  the  sum  of  the 
squares  of  the  deviations.  Find  the  median  of  3,  3,  4,  5, 
10:  and  the  sum  of  the  squares  of  the  deviations  from 
the  median.  Find  the  sum  of  the  squares  of  the  devia- 
tions from  the  harmonic  mean  (call  it  4.1)  or  from  the 
mode,  and  notice  that  S(t;2)  is  smaller  when  the  deviations 
are  measured  from  the  average  than  when  measured 
from  any  of  the  other  numerical  values. 

217.  Least  Squares  for  Conditioned  Measurements.— 
If  we  are  dealing  with  two  conditioned  measurements, 
as  in  the  case  of  the  ^-values  and  the  ^/-values  of  the 
black-thread  experiment,  the  principle  of  least  squares 

*  This  principle  can  be  proved  from  the  "fact  of  experience" 
that  deviations  follow  the  law  of  the  exponential  equation  y  =  e~xZ. 
As  an  example  of  its  application  consider  the  marks  6,  8,  and  7, 
on  a  scale  of  ten,  which  one  student  obtained  on  three  examination 
questions,  and  the  marks  7,  7,  7,  which  were  obtained  by  another 
student.  Find  the  deviations  from  the  theoretical  mark  10,  and 
see  which  student  has  the  smaller  value  for 

240 


XX 


LEAST  SQUARES 


241 


shows  that  the  line  which  expresses  the  condition  or  gives 
the  law  of  relationship  between  the  two  variables  must 
be  so  placed  that  the  sum  of  the  squares  of  the  distances 
from  it  to  all  of  the  experimental  points  shall  have  the 
smallest  possible  value. 

The  x  and  y  of  any  one  of  the  points  cannot  in  general 
be  substituted  in  the  black-thread  equation,  y  =  a  +  bx, 
but  a  +  bx  —  y  will  have 
some  small  positive  or 
negative  value  instead  of 
being  equal  to  zero.  If 
the  various  points  are 
considered  to  have  the 
definite  positions  de- 
noted by  (xi,  2/1)  >  (#2, 
1/2) ,  (#3,  2/3),  etc.,  it  will 
usually  be  found  that 
none  of  these  sets  of 
values  will  exactly  satisfy 
the  equation  y  =  a  +  bx 
or  a  -f  bx  —  y  =  0,  but 
will  give  such  a  result 
as  a  +  6x1  —  2/1  .—  di, 
where  d  is  some  small 
quantity  whose  exact 
value  need  not  be 
determined ;  similarly, 
the  other  points  will 
give  other  equations, 

a  -f  bx2  —  2/2  =  d2,  a  +  bx3  —  2/3  =  d%,  etc.,  and  ac- 
cording to  the  principle  of  least  squares  the  sum 
di2  +  dz2  +  d32  +  •  •  •  j  must  be  as  small  as  possible.* 

*  It  can  be  shown  mathematically  that  the  distance  from  the 
point  (xi,  y\)  to  the  line  y  =  a  +  bx  is  proportional  to  a  +  bxt  —  y\t 
17 


FIG.  58.  THE  PRINCIPLE  OF 
LEAST  SQUARES. — The  general  prin- 
ciple is  that  the  theoretical  rela- 
tionship is  to  be  so  arranged  that 
the  sum  of  the  squares  of  the  dis- 
crepancies of  the  actual  measure- 
ments shall  be  as  small  as  possible. 
If  a  straight  line  is  to  be  used  it 
must  be  so  arranged  that  the  sum  of 
the  squares  of  the  distances  to  it  from 
the  various  points  is  a  minimum. 


242  THEORY  OF  MEASUREMENTS  §217 

This  means  that  the  sum  of  the  squares  of  the  left-hand 
members  of  the  equations,  or  S(a  +  bxn  —  yn)2  must 
have  its  minimum  value,  and  it  can  be  shown  by  processes 
of  pure  mathematics  that  this  will  be  the  case  if 


= 

and 

= 


where  x  and  y  stand  for  Xi,  z2,  x3,  •  •  •  and  3/1,  i/2,  t/3, 
the  #-  values  and  y-  values  of  the  experimental  points. 

These  equations  enable  one  to  determine  the  values 
of  a  and  b,  and  hence  to  find  the  position  (see  §  105) 
that  a  straight  line  (black  thread)  must  have  if  it  is  to 
be  so  located  that  the  sum  of  the  squares  of  the  distances 
to  it  from  all  of  the  experimental  points  shall  have  the 
smallest  possible  value. 

By  similar  processes  a,  b,  and  c  could  be  found  for 
the  equation  of  the  parabola  y  =  a  +  bx  +  ex2,  or  the 
appropriate  coefficients  for  curves  having  even  more 
complicated  equations,  but  the  processes  of  computation 
become  so  tedious  that  it  is  better  to  replace  the  vari-. 
ables,  as  explained  in  the  lesson  on  graphic  analysis, 
by  others  that  will  conform  to  the  straight  line  law. 
The  diagram  (Fig.  58)  shows  some  of  the  points  that 
correspond  to  the  table  in  §  118,  for  which  the  position 
of  the  line  through  the  origin  was  found  graphically  by 
the  use  of  a  black  thread. 

Tabulate  the  values  of  x  and  y  for  the  black-thread 
experiment  of  §  115,  arranging  them  as  shown  in  the 
following  table,  and  writing  the  proper  numerical  values 
in  the  spaces  marked  Sx,  2y,  2(xy),  and  S(:c2).  Then 
calculate  the  values  of  a  and  b  from  the  formulae,  arrang- 


XX 


LEAST  SQUARES 


243 


ing  the  work  neatly  and  being  careful  to  avoid  using 
the  wrong  algebraical  signs  or 
confusing    S(z2)    with    (2x)2. 
Keep    only   three   significant 
figures  in  the  final  results. 

Write  the  equation  repre- 
senting the  best  position  of 
the  black  thread  in  the  form 
y  =  a  +  bx,  and  then  in  the 
form  x/m  +  y/n  =  1.  Com- 
pare the  calculated  values  of 
the  intercepts  with  your  ex- 
perimental values  that  were 
obtained  in  the  work  on 
Graphic  Analysis  (chap.  x). 

218.  Least  Squares  and  Pro- 
portionality.— If  two  variables, 
x  and  y,  are  always  propor- 
tional the  linear  equation   y  =  a  +  bx  reduces   to  the 
form 

y  =  0  +  bx 

and  the  best  value  of  the  coefficient  will  be  found  to  be 


b  = 


X 

y 

xy 

x2 

1 

9.8 

9.8 

1 

2 

8.5 

17.0 

4 

3 

8.0 

24.0 

9 

4 

7.2 

28.8 

16 

5 

6.7 

33.5 

6 

6.5 

7 

6.2 

8 

5.5 

9 

5.0 

10 

4.1 

11 

3.9 

12 

3.2 

13 

2.3 

Sw 

2(sw) 

S(z2) 

METHOD  OF  LEAST  SQUAEES 
FOR  A  LINEAR  LAW. 


Let  a  =  0  in  the  equation 


then  • 


Multiplying  by  nS(x), 


244  THEORY  OF  MEASUREMENTS  §219 

whence 


But  if  a  :  b  ::  c  :  d,  then  a  —  c  :  b  —  d  ::  c  :  d;    accord- 
ingly, 


or 


In  studying  the  density  of  water  (§  118)  the  ratio  of 
V  y  to  a:  was  found  graphically  to  be  about  0.83.  Deter- 
mine the  accurate  value  of  this  ratio  by  calculating  each 
product  of  the  variables  x  and  ^y,  summating,  and 
dividing  by  the  summated  squares  of  the  x's. 

219.  Least  Squares  for  a  Theoretically  Constant  Value. 
—  If  the  variation  of  y  is  supposed  to  be  nil,  i.  e.,  if  y  is 
a  constant,  the  best  representative  value  for  the  fluctuat- 
ing experimental  determinations  of  it  can  easily  be  found 
by  the  method  of  least  squares: 

In  the  equation  y  =  a-\-bxiib  =  Q  the  equation 

z     -  nx 
gives 

S(x)S(y)  =  nL(xy). 

Eliminating  *2,(xy)  between  this  equation  and  the  equa- 
tion 


gives 

=  ( 


which  is  obviously  the  same  as 


XX  LEAST  SQUARES  245 

In  other  words,  the  best  representative  value  of  y  is 
obtained  by  adding  all  n  of  its  experimental  values  and 
dividing  by  n, — as  already  stated  without  proof.* 

220.  Consecutive  Equal  Intervals. — If  the  successive 
unknown  intervals  of  a  scale  are  not  perceptibly  different 
from  one  another  their  value  can  easily  be  found  by  the 
method  of  coincidences,  as  illustrated  in  Fig.  48,  §  153. 
If  the  scale  is  a  crude  one  or  the  method  of  measuring  is 
a  precise  one  the  intervals  which  ought  to  be  equal  will 
be  found  to  differ  among  themselves  (§  160)  and  the 
question  arises  as  to  the  best  representative  value  for 
them.  The  principle,  of  course,  is  to  find  the  perfectly 
uniform  scale  whose  intervals  are  of  such  size  as  to  make 
the  sum  of  the  squares  of  the  discrepancies  in  the  gradua- 
tions of  the  irregular  scale  as  small  as  possible  (Fig.  59). 
It  will  not  do  to  take  the  average  of  the  lengths  of  all 
the  consecutive  intervals  because  the  result  that  would 
be  obtained  would  depend  only  upon  the  position  of 
the  first  graduation  and  the  last  graduation,  and  the 
data  afforded  by  all  the  rest  of  the  graduations  would 
have  been  neglected. 


0 

5                                                         10 

1 

1           1           I           1           1 

I             \          '<            '             *            \ 

Trt, 

Wj    1    YV»3    [ 

1       111114 

i,       !      \      \      \      \ 

1               | 

II                                   II 

3 

~l                           A  ' 

FIG.  59.  BEST  VALUE  FOR  CONSECUTIVE  INTERVALS. — An  imag- 
inary uniform  scale  is  to  be  so  chosen  that  2(i>2)  for  the  irregular 
intervals  is  a  minimum.  A,  actual  scale;  /,  7,  imaginary  scales. 

*  Remember  that  the  principle  of  least  squares  is  based  on  the 
assumption  Ithat  all  of  the  measurements  under  consideration  are 
equally  trustworthy.  If  this  condition  could  always  be  fulfilled 
we  should  have  little  use  for  any  representative  value  except  the 
average  (c/.  §§  174;  204). 


246 


THEORY  OF  MEASUREMENTS 


§220 


If  the  distance  from  the  zero  of  the  scale  to  the  gradua- 
tion that  is  marked  x  is  called  y  the  problem  is  to  find  the 
best  value  of  b  in  the  equation  y  =  bx  (Fig.  60).  The 
formula  b  =  2(:n/)/S(z2)  of  §  218  cannot  be  used  in  this 
case,  for  the  experimental  deviations  cannot  involve 

both  x  and  y,  but  must 
be  limited  to  the  y-  values 
on  account  of  the  neces- 
sity of  keeping  the  differ- 
ence between  successive 
x-values  constant.  It  is 
not  the  squares  of  the 
perpendicular  distances 
in  Fig.  58  whose  sum 
must  be  a  minimum,  but 

the  squares  of  the  vertical 
,.  .  •    **     «A    rrn 

distances  in  Fig.  60.   The 

formula    that    is    com- 
monly used  in  such  cases  can  be  obtained  as  follows  : 

If  the  n  ^-values  of  the  n  points  in  Fig.  60  are  Xi  =  0, 
z2  =  1,  x3  =  2,  •  •  •,  xn  =  n  —  1,  the  slope  6  which  will 
give  the  best  value,  m,  of  the  n.  —  1  intervals,  mi,  w2, 
ms,  -  •  •  ,  can  be  calculated  as  follows  : 


FOR  EQUAL  INTERVALS. 


Since 


+n  = 


and 


=  O2  +  I2  +  22  +  •  •  •  +  n2  =  \n(n  +  l)(2n  +  1)*, 


the  formula 


*  The  correctness  of  these  formulse  may  be  taken  for  granted. 
They  are  easily  proved  b}'  the  processes  of  elementary  algebra. 


XX 

becomes 


LEAST  SQUARES 


247 


—  n(yi  +  2y2  +  3y3  +  ----  h  nyn 


I)2  -  |n2(n 

Multiplying  both  numerator  and  denominator  by  12/n 
gives 

(n  +  l)(yi  +  3/2  -J  -----  h  y«)  -  2(yt  +  2y2  H  -----  h  nyu) 
3n(n  +  I)2  -  2n(n  +  l)(2n  +"l) 


,  = 


Simplifying  the  denominator  and  multiplying  the  fraction 
by  —  I/—  1  gives 


-  (n 
3  — 


n6  —  n 
Collecting  the  coefficients  of  the  y'a, 

,       a  (1  —  n)y:  +  (3 
0  =  0  — 


Finally,    collecting   the    coefficients   of    (n  —  l),    (n  — 3), 
etc.,  gives 


(n  -  l)(y»  -  yQ  +  (n  -  3)(yn_i  -  pp  +  (n  -  5)(yn_2  -  y»)  + 


for  the  best  value  of  the  interval.     The  point  Xi  corre- 
sponds, of  course,  to  the  zero  of  the  scale  and  yi  is 


) 

i  5   i 

1 

FIG.  61.  TREATMENT  OF  UNIFORM  INTERVALS. — The  best  repre- 
sentative value,  according  to  the  formula,  is  10 (yn  —  yi)  +  8(yio— yz) 
+  6(y9  -  y3)  +  4(y8  -  y4)  +  2(y7  -  y6)  +  0(y6  -  ye),  multiplied  by 
6/n(n2  —  1).  Notice  that  an  eyen  number  of  intervals  causes  the 
middle  graduation  to  be  neglected. 


248 


THEORY  OF  MEASUREMENTS 


§221 


subdivision 

length 

1st 

160 

2d 

163 

3d 

164 

4th 

166 

5th 

165 

6th 

159 

7th 

162 

8th 

166 

9th 

165 

10th 

166 

numerically  equal  to  zero.*  It  should  be  noticed  that 
when  there  are  an  even  number  of  intervals  the  middle 
point  does  not  affect  the  formula.  Accordingly  it  is 
preferable  to  use  an  odd  number  of  intervals,  unless 
there  is  some  reason  to  the  contrary. 

The  following  numbers  were  obtained  by  measuring 
the  consecutive  intervals  between  eleven  parallel  lines 
that  were  intended  to  be  located 
one  tenth  of  a  millimetre  apart  on 
a  glass  slide  used  for  microscopic 
measurements.  The  numbers  give 
the  corresponding  distances  in  units 
of  an  arbitrary  scale  located  in  the 
eye-piece  of  the  microscope  so  as  to 
appear  superimposed  on  the  image 
of  the  object  under  examination. 
Find  the  best  representative  value 
for  these  ten  subdivisions ;  then  note 
which  subdivision  (if  any)  agrees  in 
length  with  the  determination. 

The  same  method  can  be  used  for  equal  changes  of 
length  that  are  due  to  any  uniform  cause;  for  example, 
a  spring  elongates  by  uniform  intervals  when  weights 
proportional  to  1,  2,  3,  4,  •  •  •  are  hung  on  it.  It  is  also 
applicable  to  intervals  of  time,  e.  g.,  the  period  of  a 
pendulum  or  of  the  indicating  pointer  of  a  galvanometer 
or  analytical  balance;  and,  indeed,  to  any  variable  that 
is  proportional  to  another  quantity  which  can  be  varied 
by  equal  intervals;  thus,  the  vr-disc  (§  49)  may  be  rolled 
until  any  given  point  on  its  circumference  has  touched 
the  flat  scale  at  several  equidistant  points,  or  the  area 

*  The  student  should  be  careful  to  avoid  the  common  mistake 
of  using  2/2  instead  of  y\  for  the  first  interval  of  the  scale  (Fig.  61). 
The  result  would  be  that  only  the  other  n  —  2  intervals  would  be 
available. 


EXAMPLE     OF     SCALE 
INTERVALS. 


XX  LEAST  SQUARES  249 

indicated  by  a  planimeter  may  be  read  when  the  tracing 
point  has  been  carried  around  the  periphery  once,  twice, 
thrice,  etc. 

221.  Equal   Intervals   without   Least    Squares. — The 
black-thread  method  can  of  course  be  used  for  measure- 
ments like  those  that  have  just  been  considered,  but  if 
the  uncertainty  of  judgment  that  is  necessarily  associated 
with  it  is   objectionable   a  simple   calculation   can   be 
performed  which  is  free  from  the  disadvantage  of  aver- 
aging stated  in  the  first  paragraph  of  §  220.     The  distance 
from  yi  to  2/<«+i)/2  or  2/<»+2)/2  is  measured,  also,  from  y2  to 
2/(n+3)/2  or  i/(n-i-4)/2,  etc.,  and  each  of  the  measurements  is 
divided  by  (n  —  l)/2  or  n/2,  the  quotients  being  finally 
averaged.     Thus,  for  the  scale  shown  in  Fig.  59,  one 
fifth  of  the  distance  from  the  mark  0  to  the  five-inch 
mark  is  averaged  with  one  fifth  of  the  distance  from  2  to 
6,  one  fifth  of  3  to  7,  etc.,  so  that  all  of  the  graduation 
marks  are  utilized.* 

222.  Simultaneous  Indirect  Measurements. — In  cer- 
tain kinds  of  measurement  several  quantities  (usually 
either  two  or  three)  have  to  be  determined  by  methods 
which  will  not  allow  each  to  be  measured  separately 
but  which  furnish  functional  relations  in  which  there  is 
always  more  than  one  unknown  involved.     For  example, 
two   magnets  may  have  the  strengths   of  their   poles 
determined  by  measuring  their  relative  intensities  as  a 
ratio  and  their  mutual  attraction  as  a  product.     The 
data  x/y  =  a  and  xy  =  b  are  then  sufficient  to  deter- 
mine  both   x   and   y.     Sometimes   only   one   unknown 

*  The  procedures  given  in  §  220  and  §  221  are  the  ones  that  are 
ordinarily  used  for  physical  measurements.  Whether  their  results 
are  preferable  to  those  of  the  more  general  methods  stated  pre- 
viously will  be  an  interesting  question  for  the  student  to  determine 
for  himself.  He  should  make  up  at  least  one  example  of  an  extremely 
irregular  set  of  scale-divisions. 


250  THEORY  OF  MEASUREMENTS  §222 

quantity  needs  to  be  measured,  but  cannot  be  deter- 
mined except  along  with  a  different  one.  For  example, 
by  the  use  of  an  ammeter  to  measure  the  flow  of  electric 
current  that  a  particular  battery  can  drive  through  an 
unknown  resistance  it  is  easy  to  deduce  the  numerical 
value  of  the  resistance,,  but  the  method  must  be  modified 
before  it  can  be  used  to  measure  the  resistance  of  a 
"ground,"  i.  e.,  of  the  place  where  the  current  passes 
from  the  negligibly  resistant  wire  into  the  body  of  the 
non-resistant  earth,  for  the  current  must  return  from  the 
earth  to  the  battery  through  some  other  resistant 
"ground."  When  there  are  two  such  points,  it  is  possible 
to  find  the  sum  of  their  resistances  but  not  the  value  of 
either  one  alone.  If  three  separate  connections  to  earth 
are  available,  their  resistances  may  be  called  x,  y,  and  z, 
and  after  measuring  in  turn  the  resistances  x  +  y,  y  -f  z, 
and  x  +  z  the  value  of  any  one  of  them  may  be  found 
by  solving  the  three  simultaneous  equations  that  are 
obtained.  If  there  are  four  such  points,  however,  more 
separate  and  independent  observations  are  obtainable 
than  there  are  unknown  quantities,  and  the  determina- 
tions will  need  adjustment  by  the  method  of  least 
squares.  If  the  resistances  are  called  w,  x,  y,  and  z, 
then  the  following  data  are  available: 

w  +  x  =  a  i 

x  +  y  =  b 

y  +  z  =  c 

w          -\-  y  =  d 

x          +  z  =  e 

w                  -f  z  =  f 

In  general  no  exact  solution  will  be  possible,  and  the 
problem  is  to  determine  such  a  solution  that  the  sum  of 
the  squares  of  the  deviations  of  the  observed  values  from 


XX 


LEAST  SQUARES 


251 


the  adjusted  values  that  are  given  by  the  assumed  solution 
shall  be  as  small  as  possible. 

A  graphic  illustration  of  the  effect  of  more  equations 
than  unknowns  is  given  in  Fig.  62, 
which   represents  a  set  of  experi- 
mental determinations 
3x  -  y  =  3  -\ 

y-x=2 k 
2x  -  y  =  0  J 

By  plotting  these  three  equations 
on  one  diagram  it  will  be  seen 
that  the  three  straight  lines  do  not 
pass  through  a  single  common  in- 
tersection, but  a  point  (2.4,  4.9)  can 
be  found  which  comes  fairly  close 
to  satisfying  the  three  conditions. 

The  general  method  of  obtaining 
the  best  representative  values  of 
the  unknown  quantities  can  be 
shown  by  the  application  of  the 
principles  of  the  differential  calcu- 
lus to  be  as  follows : 

1.  Multiply  each  equation  by  the 
coefficient  of  x  in  that  equation, 
and  add  the  results.  The  sum  is 
called  a  normal  equation. 

The  equations  of  Fig.  62, 


Y 

/ 

// 

7 

/ 

•1 

/ 

9 

fr 

£ 

yj 

1 

3 

y 

LI' 

9 

/ 

4 

W 

1 

# 

/£ 

0 

/ 

1 

'I 

3  X 

FIG.  62.  THREE  OB- 
SERVATIONAL EQUA- 
TIONS.— Since  all  obser- 
vations contain  errors 
the  three  equations  are 
not  consistent,  but  the 
best  point  to  represent 
the  intersection  of  the 
three  lines  can  be  found 
by  choosing  it  so  that 
the  sum  of  the  squares 
of  the  discrepancies  is 
a  minimum. 


thus  give 


3x  -  y  =  3 

-x+y  =  2 

2x  -  y  =  0, 


x  -    y  =  -  2 
x  —  2y  =        0 


I4x  - 


7. 


252  THEORY  OF  MEASUREMENTS  §222 

2.  Multiply  each  equation  by  the  coefficient  of  y  in 
that  equation,  and  add  the  results  in  order  to  obtain 
a  second  normal  equation: 

-  3z  +    y  =  -  3 

-  x+    y  =       2 

-  2x  +    y  =       0 


-  6z  +  30  =  -  1. 

3.  If  the  equations  contain  third,  fourth,  fifth,  •  •  •  nth 
unknown  quantities,  form  a  normal  equation  for  each 
one  in  the  same  way,  multiplying  each  equation  by  its 
proper  coefficient  of  the  particular  unknown  quantity 
and  adding,  thus  obtaining  n  normal  equations  in  n 
unknown  quantities. 

4.  Solve   the   normal   equations   by   any   algebraical 
process.     The  result  will  be  the  least-square  values  of 
the  unknowns.     In  the  example  under  consideration  this 
gives 

x  =    5/2, 
V  =  14/3. 

Choose  any  other  point  that  you  think  best  and  show 
that  it  gives  a  larger  value  for  S(v2)  than  this  one  does. 
Write  the  equations  in  the  form 

3x  —  y  —  3  =  vi, 

y  —  x  —  2  =  vZ) 

2x  -  y  =  vd. 

Solve  the  following  equations  by  the  method  of  least 
squares : 

x  +  2.65y  -  0.33^  =  6.21, 
x  +  2.370  +  0.122  =  3.18, 
x  +  2.25y  -  0.292  =  5.89, 
x  -  0.870  +  3.272  =  4.28, 
x  +  3.380  +  z  =  4.07. 


XX  LEAST  SQUARES  253 

Be  careful  that  each  product  is  given  the  right  sign; 
check  them  as  each  equation  is  completed.  If  squared 
paper  is  not  used  be  careful  to  keep  the  decimal  points 
under  one  another.  The  tedium  of  adding  positive  and 
negative  decimals  can  be  greatly  relieved  by  overlining 
each  negative  digit*  and  adding  en  masse. 

An  approximate  method  of  obtaining  normal  equations 
is  sometimes  used  in  order  to  avoid  the  increased  labor 
of  applying  the  above  processes,  1,  2,  3,  4,  to  observation 
equations  that  contain  decimal  fractions,  but  it  is  often 
unsatisfactory:  Make  the  coefficient  of  x  positive  in 
each  of  the  n  equations  by  multiplying  through  by  —  1 
where  necessary;  then  add  the  resultant  n  equations  to 
form  the  first  normal  equation.  Make  the  coefficients 
of  y  positive  in  the  same  way,  and  add  the  n  equations 
to  obtain  a  second  normal  equation.  Proceed  in  this 
way  until  n  normal  equations  in  n  unknown  quantities 
have  been  formed;  then  solve.  It  is  fairly  obvious 
that  this  method  will  be  the  most  satisfactory  in  cases 
where  all  the  coefficients  of  all  the  observation  equations 
are  of  approximately  the  same  size. 

*  Find  the  value  of  2.4123  —  3.6418  +  5.7827  column  by  column 
as  follows: 

2.4123 
3.6418 

5.7827 


+  4.5532 

Verify  these  also:  .874 

3.48  .454 

7.22  _-653 

6.35  2.840 

3.380 


1.87 


254  THEORY  OF  MEASUREMENTS  §222 

Apply  the  approximate  method  to  the  last  exercise  and 
compare  the  ease  of  computation  and  the  accuracy  of 
solution. 

Apply  the  approximate  method  to  the  equations  of 
Fig.  62,  and  note  one  objectionable  feature  which  it 
may  have. 

To  what  extent  are  graphic  methods  applicable  to  the 
solution  of  simultaneous  indirect  measurements? 


XXI.     INDIRECT   MEASUREMENTS 

Apparatus. — Slide  rule. 

223.  Importance  of  Indirect  Measurements. — An  in- 
direct measurement  is  one  that  is  calculated  from  one  or 
more  direct  measurements  instead  of  being  directly  ob- 
served by  the  experimenter. 

In  a  certain  sense  almost  all  measurements  are  indirect, 
especially  those  that  require  the  highest  degrees  of  accu- 
racy. In  making  a  careful  determination  of  the  weight 
of  an  object  the  direct  determinations  are  of  the  turning 
points  of  the  pointer  that  swings  back  and  forth  as  the 
beam  of  the  balance  oscillates.  From  these  points  the 
resting  position  of  equilibrium  is  calculated,  and  from 
this  position  and  a  similar  one  obtained  when  the  balance 
is  empty  a  calculation  is  made  of  the  discrepancy  be- 
tween the  weight  of  the  object  and  that  of  the  standard 
weights  that  are  balanced  against  it,  using  a  previous 
determination  of  the  shift  of  resting  point  that  is  caused 
by  a  very  small  standard  weight.  If  the  process  of 
weighing  is  for  the  purpose  of  obtaining  mass  instead  of 
weight  still  further  calculating  is  necessary. 

The  term,  indirect,  however,  is  commonly  applied  to 
measurements  like  that  of  the  density  obtained  by 
dividing  mass  by  volume,  one  or  more  carefully  deter- 
mined measurements  being  treated  by  calculation  in 
such  a  way  as  to  give  a  required  " indirect"  result. 

In  some  cases  values  calculated  from  other  data  are  a 
necessity.  The  density  of  a  solid  may  be  obtained  by 
direct  measurement  if  it  can  be  tested  by  immersing  it 
in  fluids  whose  densities  are  known,  and  this  method  is 
found  to  be  very  convenient  for  testing  small  objects 

255 


256  THEORY  OF  MEASUREMENTS  §223 

that  do  not  have  a  very  great  density,  such  as  precious 
stones.*  For  denser  objects,  however,  some  indirect 
method  is  necessary.  The  usual  one  is  to  calculate 
d  =  m/v,  v  also  being  obtained  indirectly  from  the  loss 
of  weight  on  immersion  in  a  fluid  of  known  density. 

In  some  cases  much  time  and  laborious  calculation 
can  be  saved  by  calculating  the  average  and  probable 
error,  a  ±  dav,  for  each  quantity  that  is  measured  directly 
and  then  investigating  the  resultant  probable  error  of 
the  indirect  measurement  in  accordance  with  theoretical 
considerations.  For  example,  it  would  take  much  longer 
to  calculate  a  density  from  each  one  of  a  set  of  twenty-five 
determinations  of  mass  and  volume  and  then  average 
the  results  than  it  would  to  average  the  masses  and  the 
volumes  separately  and  perform  a  single  division.  When 
using  the  latter  method  it  is  possible  to  find  what  the 
probable  error  of  the  density  will  amount  to  by  investi- 
gating the  probable  errors  of  mass  and  volume. 

In  a  case  like  the  one  just  mentioned  it  would  also 
be  possible  to  make  twenty-five  calculations  of  density 
from  the  twenty-five  sets  of  measurements  of  mass  and 
density  and  to  compute  the  probable  error  of  the 
average  density.  In  many  cases,  however,  such  a  pro- 
cess cannot  be  carried  out.  If  there  were  more  data  at 
hand  for  the  mass  than  for  the  volume  the  excess  could 
not  be  utilized;  and,  furthermore,  some  of  the  values 
needed  in  the  formula  may  be  predetermined  constants, 
such  as  those  mentioned  in  chapter  xviii  (see  §  197),  which 
are  given  only  in  the  form  of  a  representative  magnitude 
and  its  probable  error.  The  question  then  arises  as  to 

*  A  solution  of  mercuric  potassium  iodide,  sp.  gr.  =  3.11  or  less, 
is  used  by  jewelers  for  this  purpose.  It  is  stated  that  densities  as 
high  as  3.56  can  be  determined  by  using  the  double  iodide  of  barium 
and  mercury. 


XXI  INDIRECT  MEASUREMENTS  257 

the  way  in  which  the  probable  error  of  the  indirect 
measurement  is  influenced  by  the  size  of  the  probable 
errors  of  the  direct  measurements  on  which  it  is  based. 

224.  Probable  Error  of  a  Sum. — The  simplest  case  is 
when  the  indirect  measurement  is  merely  the  sum  of  the 
two  independent  direct  measurements.     Let  the  direct 
measurements  be  denoted  by  the  small  letters, 

ai  db  Pi    and     a2  ±  pz, 

when  stated  in  the  form  of  average  and  probable  error, 
and  let  the  indirect  measurement  with  its  probable  error 
be  represented  by  capitals, 

A±P, 

the  direct  measurement  being  equal  to  the  sum  of  the 
indirect  measurements,  so  that  A  =  ai  +  a2;  then  it  can 
be  proved  that 

P2  =  Pi2  +  P/. 

The  dispersion  of  the  sum  is  not  as  large  as  the  sum  of 
the  two  direct  dispersions  from  which  it  is  calculated. 
This  is  because  positive  and  negative  deviations  will 
counterbalance  each  other  to  a  certain  extent.  It  is 
larger  than  either  of  the  others  alone,  however,  for  the 
extra  measurement  gives  an  extra  degree  of  uncertainty. 
A  bench  has  a  height  of  42.50  ±  .03  cm.  above  the 
floor,  and  a  table  is  44.35  d=  .04  cm.  higher  than  the 
bench.  Write  the  height  of  the  table,  with  its  probable 
error. 

225.  Probable  Error  of  a  Difference. — If  the  indirect 
measurement,   A,   is   equal  to   the   difference,   a\  —  a2, 
between  two  direct  measurements,  0,1  and  a2,  it  is  perhaps 
a  natural  inference  that  the  square  of  its  dispersion,  P2, 

18 


258  THEORY  OF  MEASUREMENTS  §226 

should  be  equal  to  pi2  —  p22.     This  is  not  the  case,  how- 
ever, but 

P2  =  Pi2  +  ?22 
in  all  cases  in  which 

A  =  «i  ±  CL. 

The  table  that  was  measured  in  the  last  exercise  is 
51.10  ±  .04  cm.  lower  than  a  shelf  which  is  137.95  ±  .03 
cm.  above  the  floor.  How  high  is  the  table? 

Notice  that  the  same  formula  applies  to  both  of  these 
exercises.  The  difference  of  two  measurements  has  as 
large  a  probable  error  as  their  sum.  Measuring  first  up 
and  then  down  does  not  give  a  more  precise  result  than 
measuring  up  and  then  further  up. 

226.  Probable  Error  of  a  Multiple.—  If 

A  =  citti, 

where  Ci  is  some  constant  not  subject  to  error,  then 
P2  =  ci  or     P  = 


The  diameter  of  a  circular  disc  is  found  to  be  7.98  ±  .03 
cm.  What  is  its  circumference? 

Notice  that  the  relative  dispersion  of  the  circumference 
(3/800)  is  the  same  as  the  relative  dispersion  of  the 
diameter  (vide  infra). 

227.  Associative  Law:— 

In  general,  if 

A  =  Ciai  db  c2a2  ±  c3a3  ±  •  •  • 
then 

P2    =    dV   +   C22P22    +   C32P32    +    •  -  •  (1) 


where  pn  is  the  probable  error  of  the  average  an. 

This  formula  can  be  used  to  find  the  probable  error 
of  an  algebraical  expression  when  the  probable  error  of 
each  of  its  terms  is  known. 


XXI  INDIRECT  MEASUREMENTS  259 

A  wall  consists  of  15  courses  of  bricks,  each  of  which  is 
56.5  =t  .5  mm.  thick,  separated  by  14  layers  of  mortar 
which  have  an  average  thickness  of  7.5  ±1.5  mm.  Show 
that  the  probable  error  of  the  height  of  the  wall  is  ±  22.3 
mm.,  and  state  how  much  this  figure  would  be  reduced 
if  the  bricks  were  absolutely  uniform.  How  much  if  the 
mortar  was  of  uniform  thickness  instead? 

228.  Probable  Error  of  a  Product.  —  If  two  independ- 
ent measurements  are  multiplied  together  the  probable 
error  of  the  product  will  follow  the  law  expressed  in  the 
following  equation. 
If 

A  =  ai&2 
then 

P2  =  piW  +  aiV- 
Likewise  if 

A  = 
then 

P2  = 


and  similarly  for  any  number  of  factors;  but  it  is  more 
satisfactory  in  practice  to  make  use  of  the  relative  prob- 
able error  (relative  dispersion,  §  195)  as  in  the  following 
form,  which  is  easily  deducible  from  the  form  just  given. 
If 

A  =  a\  «2  «3  •  •  • 
then 

(P/A)2  =  (Pl/atf  +  (p2/a2)2  +  (p3/a3)2  +  -  •  •. 

Prove  that  the  last  equation   (as  far  as  it  goes)  is 
equivalent  to 

P2  =  (pic^as)2  +  (aip2a3)2  +  (aia2p3)2. 


A  rectangular  block  measures  20.00  db  .04  cm.  in 
length,  10.00  =t  .01  cm.  in  breadth,  and  5.00  ±  .01  cm. 
in  thickness.  What  is  its  volume? 


260  THEORY  OF  MEASUREMENTS  §230 

229.  Probable  Error  of  a  Power.— If 

A  =  «in 
then 

(P/A)2  =  (rcW/arO     or     PyA 

and,  likewise,  if 

A  = 
then 

P/A  = 

the  constant  not  appearing  in  the  formula  if  the  relative 
probable  error  is  used. 

In  the  particular  case  for  which  n  =  1,  notice  that  the 
relative  probable  error  of  Cidi  is  the  same  as  the  relative 
probable  error  of  a\  itself  (vide  supra). 

One  edge  of  a  cubical  block  is  10.00  ±  .01  cm.  What 
is  its  volume?  What  is  the  area  of  its  total  surface? 

Does  the  rectangular  block  (above)  appear  to  have 
been  measured  as  accurately  as  the  cubical  block?  How 
do  their  volumes  compare  in  respect  to  accuracy?  What 
is  there  about  the  data  of  the  rectangular  block  which 
make  its  volume  determinable  as  closely  as  that  of  the 
cubical  block,  although  its  measurements  are  less  precise? 

Show  that  the  combined  volume  of  the  two  blocks  is 
2000.  =t  4.2  cm3. 

230.  Distributive  Law :  — 

In  general,  whether  the  exponents  are  positive,  nega- 
tive, or  fractional,  if 

A  =  Ciaima2na3r  •  •  • 
then 

(P/A)2  =  (mPl/ai)2  +  (np2/a2)2  +  (rp3/a3)2  +  -  •  -.     (2) 

This  formula  can  be  used  to  find  the  probable  error 
of  any  single  term  of  an  algebraical  expression  when 
the  probable  errors  of  its  factors  are  known. 


XXI  INDIRECT  MEASUREMENTS  261 

The  formula  for  the  volume  of  a  cylinder  is 
v  =  irlr2. 

If  the  measurements  of  I  and  r  have  respective  probable 
errors  of  pt  and  pr  find  the  value  of  pv,  the  probable 
error  of  the  calculated  volume. 

Test  the  accuracy  of  the  italicized  statement  (that 
dav  =  di/^n)  of  §  193  by  letting  p  denote  the  probable 
error  of  each  of  the  n  single  measurements  and  finding 
the  probable  error  of  their  average. 

231.  Recapitulation. — In  the  following  formulae  the 
relative  dispersion  (P/A,  or  p/a)  is  represented  by  R  or  r. 

If  Then 


A  =  o 

i  rb  02  =b   •  •  • 

P2   =  Pi2  +  P22  +   '  •  ' 

A  =  c 

itti 

P     =  Cip! 

A  =  c 

lOi  zt  Ctttz  =t   •  '  ' 

P2   =    (Ci^i)2  +   (C2p2)2  +    •  •  ' 

PROBABLE  ERRORS  OF  INDIRECT  MEASUREMENTS. — In  these 
cases  the  probable  error  (p)  for  each  average  (a)  is  most  convenient 
for  calculation.  The  capital  letters  refer  to  the  indirect  measure- 
ment. 


// 

TAen 

A  =  Oin 
A  =  CiOin 

R2  =  r!2  - 
R    =  nr 
R    =  nr 
R2  =  (mr 

O2  +  (nr2)2  H 

-  (srs)2  +  -  •  • 

PROBABLE  ERRORS  OF  INDIRECT  MEASUREMENTS. — In  these 
cases  the  use  of  the  relative  probable  error  (r)  simplifies  the  calcula- 
tion. 

Notice  the  complete  formal  correspondence  between 
(absolute)  probable  errors  after  adding,  multiplying  by  a 
coefficient,  and  assembling  terms,  and  relative  probable 
errors  after  multiplying,  raising  to  a  power,  and  assem- 
bling factors,  respectively. 


262  THEORY  OF  MEASUREMENTS  §232 

A  bowl  whose  interior  is  an  exact  segment  of  a  sphere 
is  found  to  have  a  depth  of  25.00  ±  .02  centimeters 
and  a  diameter  across  the  top  of  50.00  ±  .30  centimeters. 
Find  its  capacity  from  the  formula  for  the  volume  of  a 
spherical  segment,  v  =  irhr2/2  +  Trhs/6,  where  h  is  the 
height  or  depth  of  the  segment  and  r  is  the  radius  of  its 
circular  base;  find  the  probable  error  of  the  capacity 
by  applying  the  second  general  equation  to  each  term 
of  the  formula  and  then  using  the  first  general  equation 
to  determine  the  final  result.  Notice  the  relative  prob- 
able error  of  the  radius,  r,  is  the  same  as  that  of  the 
diameter,  d.  Arrange  the  calculation  systematically  in 
order  to  avoid  numerical  mistakes,  and  if  there  is  any 
trouble  in  making  the  substitution  write  out  each  step 
of  the  process;  for  example: 

ai  =  25.       P2/A2  =  32(.02)2/252  log  25  -  1.3979 

pi  =  .02  P2  =  32(.02)27r2256/25262     log  7r2       0.9943 

m  =  3  =  7r2(.01)2254  4.0000 

d  =  7T/6  5.5916 

A  =  7r/*3/6  2.5859 

logP  = 

232.  Graphs  of  Propagated  Errors. — It  has  been  seen 
that  the  probable  errors  of  two  or  more  direct  measure- 
ments are  propagated  through  any  kind  of  a  calculation 
and  give  the  indirect  measurement  a  probable  error 
whose  formula  is  of  the  type  Vz2  +  y2.  Since  the  square 
root  of  the  sum  of  two  squares  can  always  be  represented 
by  the  hypothenuse  of  a  right-angled  triangle  a  graphic 
solution  of  the  probable  error  of  an  indirect  measurement 
is  easily  effected. 

From  any  "origin"  draw  a  horizontal  line  and  a 
vertical  line,  making  their  lengths  equal  to  the  disper- 
sions 3  and  4  of  the  measurements  in  §  225.  Complete 


XXI 


INDIRECT  MEASUREMENTS 


263 


the  right-angled  triangle  and  note  that  the  length  of  the 
hypothenuse,  5,  is  the  " propagated"  dispersion  of  the 
indirect  measurement,  as  pre- 
viously found  by  calculation.* 
The  fact  that  pi2  +  p22  + 
p32    is    the    same    thing    as 


V 


FIG.  63.  GEOMETRIC  DE- 
TERMINATION OF  PROPA- 
GATED ERRORS. 


Vpi2  +  p  +  Ps2,  namely,  the 
sum  of  two  squares,  makes 
this  method  extensible  to  any 
number  of  terms  (Fig.  63). 

Use  the  geometrical  con- 
struction to  find  the  square 
root  of  the  sum  of  the  squares 

of  4,  3,  and  12.     Afterward,  verify  your  result  by  calcu- 
lation. 

233.  Relative  Importance  of  Compound  Errors.  —  The 
fact  that  an  indirect  probable  error  which  depends  upon 
the  measurement  of  two  or  more  different  quantities 
always  assumes  the  form  Vx2  +  y2  means  that  it  will 
be  more  decidedly  diminished  by  reducing  the  larger  of 
the  two  independent  probable  errors  than  by  attempting 
to  improve  the  more  accurate  measurement.  Show  that 
A/52  -f  22  is  reduced  by  41%  if  the  5  is  changed  to  2.5, 
but  only  by  5%  if  the  2  is  changed  to  1. 

Draw  a  right-angled  triangle  with  one  side  several 
(say  8  or  10)  times  as  long  as  the  other.  Change  the 
long  side  by  making  it  a  little  longer  or  shorter  and 
notice  that  the  change  in  length  of  the  hypothenuse 
is  almost  in  exact  proportion.  Change  the  short  side 
and  notice  that  the  hypothenuse  is  hardly  affected  at  all. 

The  formula  for  the  volume  of  a  cylinder  is  v  = 


*  These  numbers  refer,  of  course,  to  hundredths  cf  a  centimetre. 
It  would  be  possible,  but  .not  advisable,  to  perform  the  operation 
A/.042  +  .032  =  .05  so  as  to  obtain  the  answer  in  centimetres  (188). 


264  THEORY  OF  MEASUREMENTS  §234 

In  determining  this  indirect  measurement  which  of  the 
two  dimensions  ought  to  be  measured  the  more  carefully? 
How  much  more  carefully?  Why? 

234.  Questions  and  Exercises. — 1.  Devise  a  method 
for  determining  velocities  by  direct  measurement. 

2.  Each  volume  of  a  ten-volume  encyclopedia  has 
5.0  cm.  ±  .1  cm.  thickness  of  leaves  between  its  two 
covers.  Each  cover  has  a  thickness  of  .35  cm.  db  .02  cm. 
How  much  more  shelf-room  than  57.00  cm.  is  it  likely 
to  need?  What  is  the  meaning  of  the  word  " likely"  in 
the  last  sentence? 


XXII.     SYSTEMATIC  AND  CONSTANT  ERRORS 

Apparatus. — Clock,  chronometer,  or  time  circuit,  giv- 
ing audible  seconds ;  watch  with  second  hand;  slide  rule. 

235.  Definitions. — It  has  been  shown  that  errors  may 
be  either  accidental  or  constant.     There  is  another  class 
of  errors,  often  included  under  the  term  constant  errors, 
in  which  the  error  is  not  actually  constant,  nor  does  it 
vary  according  to  the  law  of  probability.     This  is  the 
class  of  systematic  errors,  or  errors  that  undergo  a  more 
or  less  regular  change  during  the  course  of  making  a  set 
of  measurements.     They  may  be  subdivided  into  pro- 
gressive errors,  which  show  a  steady  increase  (or  decrease) 
from  one  determination  to  the  next,  and  periodic  errors, 
which   increase   for   a   number   of   measurements,   then 
decrease,  and  then  repeat  the  previous  cycle  or  period. 

236.  A  Test  for  Systematic  Errors. — Where  systematic 
errors  are  absent  a  comparison  of  any  measurement  of  a 
series  with  the  preceding  one  will  tend  to  show  an  increase 
in  the  numerical  value  about  as  often  as  a  decrease;    a 
fact  that  can  easily  be  tested  by  writing  between  each 
two  successive  values  a  plus  sign,  a  minus  sign,  or  a  zero, 
according  as  the  second  value  is  respectively  greater 
than,  less  than,  or  equal  to,  the  first,  and  then  comparing 
the  number  of  the  plus  signs  with  that  of  the  minus  signs. 

Where  progressive  errors  have  been  greater  than  acci- 
dental errors  there  may  be  all  plus  signs  or  all  minus 
signs  as  the  result  of  applying  the  test.  If  the  accidental 
errors  are  relatively  large  they  will  probably  cause  several 
of  the  signs  to  be  plus  and  several  minus,  but  the  presence 
of  a  progressive  error  at  the  same  time  will  cause  one 
sign  to  appear  more  frequently  than  the  other. 

265 


266  THEORY  OF  MEASUREMENTS  §238 

If  the  systematic  errors  are  periodic  there  will  be 
alternate  groups  of  plus  signs  and  minus  signs,  as  is 
shown  in  the  next  table. 

237.  Example  of  a  Systematic  Error. — In  an  experi- 
ment in  which  water  in  a  reservoir  was  drawn  up  into 
a  tube  by  suction  and  successive  readings  of  its  height 
were  made  values  having  the  following  decimals  were 
obtained  in  order:    .76,  .74,  ,70,  .62,  .63,  .61,  .55,  .56, 
.51,  .50,  .44,  .44,  .39,  .40,  .35.  .35.     Are  the  results  prob- 
ably affected  by  progressive  errors,  or  periodic  errors,  or 
neither?     Use  a  graphic  diagram  if  the  question  is  hard 
to  answer.     What  effect  would  you  expect  to  result  from 
a  slight  leakage  of  air  into  the  upper  part  of  the  tube? 

238.  Example  of  a  Periodic  Error. — If  the  pivot  of  the 
second  hand  of  a  watch  is  not  exactly  in  the  center  of 
the  dial  the  indicated  seconds  will  be  subject  to  a  periodic 
error.     For  example,  if  it  is  located  too  far  to  the  right 
the  hand  may  indicate  29  instead  of  30  when  it  points 
downward,  and  1  instead  of  60  when  it  is  pointing  up- 
ward.    That  is,  it  will  have  a  periodic  error  which  will 
be  a  maximum   (positive)   at  60  seconds,   a  minimum 
(negative)  at  30  seconds,  and  zero  at  15  and  45  seconds 
(Fig.  64).     From  the  illustration  it  is  evident  that  the 
direction  of  displacement  must  be  toward  a  point  half- 
way between  the  positions  at  which  the  error  is  greatest 
and  least. 

Stand  where  you  can  hear  the  clock  beat  seconds  and 
read  the  time  indicated  by  your  watch.  Every  seven 
seconds  as  indicated  by  the  clock  read  the  seconds  and 
estimated  tenths  of  a  second  from  the  watch  and  state 
the  result  to  another  student,  who  will  take  down  the 
values  in  his  notebook.  After  three  or  four  minutes 
change  places  with  him  and  note  down  the  time  as  he 
reads  it  off.  Every  seven-second  interval  should  have 


XXII       SYSTEMATIC  AND  CONSTANT  ERRORS 


267 


its  time  by  the  watch  noted,  for  a  full  period  of  seven 
minutes.  It  is  advisable  to  practice  the  procedure  be- 
forehand until  you  are  sure  that  you  can  estimate  tenths 
of  a  second  with  reasonable  accuracy.  If  your  estimates 


YIO  error 


no  error 


FIG.  64.  ILLUSTRATION  OF  A  PERIODIC  ERROR. — An  eccentric 
clock-hand  will  appear  to  be  ahead  of  time,  accurate,  behind, 
accurate,  ahead,  etc.,  in  the  course  of  its  rotation.  A  determination 
of  the  times  at  which  the  gain  and  the  loss  are  most  marked  will 
enable  the  direction  and  amount  of  displacement  to  be  found. 

are  predominantly  0.5  and  0.0  the  results  will  not  be 
satisfactory.  In  order  to  avoid  losing  count  of  the 
(audible)  seconds  while  you  are  stating  the  time  it  may 
be  advantageous  to  choose  seven  of  your  fingers  and  tap 
on  the  table  with  each  in  turn  at  one-second  intervals. 
Concentrate  your  sense  of  sight  on  the  watch  and  your 
sense  of  hearing  on  the  beats  of  the  clock. 

See  that  you  have  the  complete  table  of  sixty  values 
in  your  own  notebook,  and  mark  the  observed  tenths  of 
a  second  with  a  plus  sign  where  they  increase  from  one 
observation  to  the  next  and  with  a  minus  sign  where 
they  decrease,  as  shown  below.  With  most  watches  it 


268  THEORY  OF  MEASUREMENTS  §238 

will  be  found  that  the  second  hand  is  not  pivoted  in  the 
exact  centre  of  the  graduated  circle  and  the  periodic 
error  will  be  shown  very  distinctly. 

hr.min.sec.  hr.min.sec.  hr.min.sec. 

4:42:45.2 
52.0, 
59.11 
43:06.31 
13.51 
20.61 
27.8  + 
34.6  _ 
41.3 
48.1  _ 
55.0  , 


4:37:65.2  , 

4:40:25.8 

38:12.31 

32.6 

19.6+ 

39.3 

26.6  u 

46.1 

33.5 

53.0  , 

40.3 

60.2  + 

47.1  0 
54.1  £ 
39:01.1  Y 

08.2  + 

41:07.4  + 
14.6  + 

21.7  + 
28.7° 

15.4  + 

35.6 

22.6  + 

42.2 

29.6  u 

49.1,. 

36.4  ~ 

56.1  £ 

43.1~ 

42:03.2  + 

50.0  ~ 

57.02 

10.3  + 
17  5 

64.2  + 

24.7  + 

40:11.4+ 

31.6" 

18.6  + 

38.4  ~ 

44:02.3 
09.4 
16.6 
23.8 
30.7 
37.4 


.^ 
51.  IX 

68.12 

4:45:05.2  + 

APPARENT  TIME  OF  SEVEN-SECOND  INTERVALS.  —  The  tabulated 
numbers  are  the  times  indicated  by  a  watch  at  audible  intervals 
that  were  known  to  be  exactly  seven  seconds.  The  presence  of  a 
periodic  error  is  shown  by  the  tabular  differences  occurring  in  alter- 
nate groups  of  positive  and  negative  values. 

Draw  a  graphic  diagram  in  which  the  abscissae  repre- 
sent the  integral  part  of  the  number  of  seconds  in  your 
table,  and  the  ordinates  represent  the  corresponding 
tenths  of  a  second  (Fig.  65).  Draw  a  smooth  curve  to 
eliminate  accidental  errors  in  the  determination  of  time. 

Determine  the  direction  and  the  amount  of  the  dis- 
placement, and  summarize  the  result  by  stating  "the 
pivot  of  the  second  hand  of  the  watch  is  displaced  toward 
the  figure  •  •  •  of  the  dial  by  an  amount  equal  to  the 
length  of  •  •  •  seconds'  divisions  on  the  graduated  circle." 


XXII       SYSTEMATIC  AND  CONSTANT  ERRORS          269 


.3  * 

02      O 

ll 

-     £ 


11 

-+->     d) 


§1 


^      ^ 

.    S    C3 


E±g 


270 


THEORY  OF  MEASUREMENTS 


§239 


i    i 
I 


Explain  how  the  periodic  error  can  be  eliminated  in 
case  such  a  watch  is  used  for  determining  intervals  of 
time. 

239.  Example  of  a  Progressive  Error.— The  list  of  fig- 
ures given  in   §  237  was  obtained 
t  |  from    a   determination    of  specific 

jLJl  gravity  by  Hare's  method.     If  the 

lower  ends  of  two  upright  tubes 
dip  into  two  separate  reservoirs 
while  their  upper  ends  are  both 
joined  to  a  third  tube  from  which 
the  air  can  be  partially  exhausted 
it  can  easily  be  proved  that  the 
heights  to  which  the  fluids  are 
raised  will  be  inversely  proportional 
to  their  densities ;  so  that  if  a  fluid 
whose  density  is  unity  is  raised  to  a 
height  hi,  and  a  heavier  fluid  to  a 
lesser  height  h%,  the  density  or  spe- 
cific gravity  of  the  latter  must  be 
hi/hz.  The  complete  list  of  deter- 
minations of  height  included  read- 
ings of  both  columns  of  liquid;  they 
were  made  at  approximately  equal 
intervals  of  time,  and  in  the  order 
in  which  they  are  given  in  the  table, 
viz.,  75.76,  73.06,  75.74,  73.04,  75.70,  73.00,  72.98,  72.95, 
75.62,  etc. 

If  the  density  is  calculated  by  dividing  75.76  by  73.06 
it  is  evident  that  the  progressive  error  will  make  the 
resulting  figure  too  large,  for  the  height  of  the  water 
had  fallen  somewhat  below  75.76  when  the  reading  of 
the  salt  solution,  73.06,  was  taken;  and  if  75.74  is  divided 
by  73.06  the  progressive  error  will  make  the  result  too 


FIG.  66.  HARE'S 
METHOD  OF  BALANC- 
ING COLUMNS.  —  The 
heights  of  the  two  liq- 
uids are  inversely  pro- 
portional to  their  den- 
sities. 


XXII       SYSTEMATIC  AND  CONSTANT  ERRORS 


271 


pure 
water 


salt 
sol  lion 


small,  for  the  salt  solution  did  not  stay  at  73.06  while  the 
reading  75.74  was  being  taken.  Obviously  the  average 
of  75.76  and  75.74  must  be  divided  by 
73.06,  or  75.74  must  be  divided  by  the 
average  of  73.06  and  73.04,  or  some 
other  combination  used  in  which  the 
average  height  of  one  column  of  liquid 
must  have  occurred  at  the  same  time 
as  the  average  height  of  the  other. 
This  method  of  eliminating  progres- 
sive errors  is  used  in  the  process  of 
weighing  with  a  delicate  balance  and  in 
many  other  processes  of  physical  meas- 
urement. 

What  set  of  values  near  the  end  of  the 
table  can  be  used  in  the  same  way? 
Make  five  different  calculations  of  density 
from  successive  parts  of  the  table  and  see 
whether  they  show  any  evidence  of  pro- 
gressive error. 

240.  Constant  Errors. — It  has  already 
been  stated  that  constant  errors  are  more 
troublesome  than  accidental  errors  and 
that  the  latter  give  very  little  aid  in  de- 
termining the  former.  It  is  not  the  tar- 
get (page  193)  that  is  found  from  indi- 
vidual measurements  but  only  the  centre 
of  clustering,  and  characteristic  deviations 
show  only  how  close  determinations  come 
to  one  another,  not  how  close  they  come  to 
the  truth. 

Some  constant  errors  are  easily  corrected  with  the  aid 
of  theoretical  considerations ;  others  may  be  very  difficult 
to  eliminate.  Unfortunately  there  is  no  infallible  rule 


75.76 


75.74 
75.70 


75.62 

.63 
.61 


.55 
.56 


.51 
.50 


.44 
.44 


.39 
.40 


.35 
.35 


73.06 
73.04 

73.00 

72.98 
.95 

.94 


.89 

.88 


.84 
.84 


.79 

.78 


.75 
.70 


.77 

.77 


EXAMPLE  OF 
HEIGHTS  IN 
HARE'S  METHOD. 


272  THEORY  OF  MEASUREMENTS  §240 

for  detecting  them,  and  each  experimental  problem  has 
its  own  special  sources  of  error.  The  two  beam-arms 
of  a  balance  may  be  unequal,  so  that  all  weighings  are 
proportionately  erroneous;  the  end  of  a  metre-stick  may 
be  worn,  so  that  every  setting  of  the  zero-point  is  in- 
accurate; the  neutral  tint  of  litmus  may  be  faultily 
judged,  so  that  a  chemical  determination  is  biassed. 
Consider  such  a  simple  process  as  the  determination  of 
atmospheric  pressure  with  a  mercurial  barometer.  The 
vacuum  at  the  top  is  never  perfect  and  there  is  often 
capillary  action,  both  making  the  reading  too  low.  If 
the  barometer  and  its  attached  scale  do  not  hang  verti- 
cally every  apparent  reading  will  be  too  high.  The 
scale  itself  is  too  long  or  too  short  except  at  a  single 
temperature,  and  the  mercury  may  have  its  accepted 
standard  density  only  at  a  different  temperature  from 
the  one  that  it  has  when  the  observation  is  made.  Even 
if  its  density  is  standard  the  height  of  a  column  that  will 
give  a  definite  pressure  will  depend  upon  the  strength 
of  gravitational  attraction  and  this  varies  with  the  lati- 
tude and  altitude  of  the  instrument.  If  an  aneroid 
barometer  is  to  be  used  instead  of  a  mercurial  one  its 
mechanism  introduces  still  more  sources  of  error. 

It  is  evident  that  the  amount  of  constant  error  will 
generally  be  varied  by  changing  observers,  apparatus, 
methods,  and  times  of  observation;  and  the  more  rad- 
ically different  the  sets  of  conditions  are  made  the  better, 
in  all  probability,  will  be  the  mutual  neutralization  of 
constant  errors  when  the  weighted  average  is  taken. 
In  practice,  the  values  for  most  of  the  constants  of 
nature  have  been  obtained  under  such  varying  condi- 
tions. Atomic  weights  are  obtained  from  various  inter- 
relations of  chemical  compounds  obtained  from  different 
sources  and  by  different  methods.  The  surface  tension 


XXII      SYSTEMATIC  AND  CONSTANT  ERRORS          273 

of  water  may  be  measured  by  the  hanging  drop  method, 
by  the  capillary  wave  method,  by  the  vibrating  jet 
method,  etc.  The  size  of  the  molecules  of  a  gas  may 
be  calculated  from  the  rate  at  which  heat  is  conducted 
through  them,  from  the  covolume  constant,  6,  of  Van  der 
Waal's  equation,  from  experimental  determinations  of 
the  viscosity  of  the  gas,  from  measurements  of  the 
maximum  density  obtainable  by  cooling  and  liquefying 
or  solidifying  it,  etc.  If  various  determinations  agree 
closely  in  spite  of  the  employment  of  essentially  different 
methods  it  becomes  more  probable  that  constant  errors 
have  been  satisfactorily  removed,  but  it  can  never  be 
certain  that  all  of  these  methods  have  not  some  common 
source  of  error  which  would  be  eliminated  only  by  using 
some  entirely  different  method.  Constant  watchfulness, 
as  stated  in  §  162,  and  the  exercise  of  good  judgment  are 
of  the  greatest  importance  in  guarding  against  constant 
errors.  If  the  student  takes  up  further  courses  that 
involve  accurate  measurement  he  will  usually  find  that 
various  " sources  of  error"  which  have  been  found  by 
previous  experimenters  will  be  explicitly  stated.  Many 
of  them  will  be  sources  of  constant  error,  and  both  his 
natural  ability  and  his  progress  in  learning  will  be  put 
to  the  test  in  his  management  of  them. 


ID 


APPENDIX 


TABLES 


EXPLANATORY  NOTES 

Formulae page  282 

Equivalents page  283 

The  logarithm  of  each  stated  factor  is  given  in  another 
column  for  convenience  of  computation.  The  table  of 
approximate  equivalents  is  for  use  when  no  great  accu- 
racy is  required. 

Greek  Alphabet page  284 

Size  of  Errors page  284 

The  words  given  in  this  table  will  help  to  fix  the 
attention  better  than  the  numerical  quantities. 

Characteristic  Deviations page  284 

General  Sources  of  Error "     284 

Density  of  Water "     285 

Notice  that  the  density  (mass  per  volume)  of  water 

(under  atmospheric  pressure)  is  never  as  great  as  unity.* 

A  parallel  column  gives  the  specific  gravity  with  reference 

to  water  at  the  temperature  of  maximum  density. 

Inverse  Tangents  and  Circular  Measure ....  page  285 

Squares  and  Square  Roots pages  286-287 

The  squares  and  square  roots  of  all  numbers  are 
obtained  with  four-figure  accuracy  by  using  this  table 
like  any  logarithm  table.  Complete  five-figure  and  six- 
figure  squares  of  all  three-figure  number  are  obtained 

*  The  terms  density  and  specific  gravity  are  often  confused,  even  in 
text  books.  Sometimes  an  arbitrary  unit  of  volume  is  substituted 
for  the  cubic  centimetre  in  order  that  the  maximum  density  of  water 
may  appear  to  be  unity,  and  a  note  is  added  to  the  effect  that  the 
density  is  stated  in  "  grams  per  millilitre."  (A  litre  is  defined  as  the 
volume  of  1000  grams  of  water  at  the  temperature  of  maximum 
density.) 

277 


278  THEORY  OF  MEASUREMENTS 

as  follows:  the  last  two  figures  of  the  required  square 
(N2)  are  printed  in  italic  opposite  the  last  two  figures 
of  N  in  the  margin.  These  can  then  be  used  without 
ambiguity  to  correct  the  four-figure  value  obtained  in  the 
ordinary  manner.  E.  g.,  required  the  square  of  417. 
The  table  gives  the  approximate  value  (pointed  off  by 
inspection)  of  173900.  Opposite  17  are  found  the  italic 
figures  89.  The  square  is  accordingly  173889;  not  173989, 
for  the  latter  would  round  off  to  174000  instead  of  17390Q. 

The  Probability  Integral .page  288 

The  tabular  value  gives  the  area  under  the  curve 
y  =  e~x2  between  the  ordinates  0  and  x  in  terms  of  the 
total  area  between  x  =  0  and  x  =  oo.  In  the  small 
table  of  y  and  z/,47694  the  value  of  I  for  Chauvenet's 
criterion  will  be  found  in  the  first  column  opposite  the 
value  of  1  —  l/2n  in  the  second  column. 

Five-Place  Logarithms page  289 

This  table  is  used  like  the  four-place  one.  The  first 
three  figures  of  the  natural  number  (antilogarithm)  are 
found  at  the  left,  the  fourth  at  the  top,  for  the  fifth  inter- 
polation is  necessary. 

If  four-figure  values  are  desired,  round  off  a  final  five 
by  increasing  the  previous  figure  unless  the  five  is  fol- 
lowed by  a  minus  sign;  e.  g.,  log  1.055  =  .0233,  but  log 
1.065  =  .0273,  not  .0274. 

Fifth  Place  of  Logarithms .  . page  289 

Before  annexing  an  italicized  figure  the  four-figure  value 
must  be  decreased  by  unity.  E.  g.,  log  839  =  92376,  not 
92386;  log  274  =  .43775,  but  log  282  =  45025. 

The  "tabular  differences"  maybe  nearly  as  large  as 
four  hundred,  but  are  easily  handled  by  logarithms  for 
purposes  of  interpolation.*  E.  g.,  to  find  antilog  22222: 

*See  Four-Place  Logarithms,  page  279. 


APPENDIX  279 

antilog  22011  =  16.;  antilog  22272  =  167;  tab.  dif.=261; 
tabular  excess  of  given  logarithm  =  211;  211/267  is  about 
0.8  (§  14)  and  is  found  by  subtracting  log  267  ( =  4265) 
from  log  211  (  =  3243),  giving  8978  (  =  log  790);  /.antilog 
22222  =  166|||  =  16d79. 

Exponentials page  289 

A  subscript  number  takes  the  place  of  a  decimal  point 
followed  by  the  corresponding  number  of  ciphers.  E.  g., 
044343  means  0.00004343,  31234  means  .0001234,  etc. 

Notice  that  ex+y  =  exev'}  for  example,  e2-5  is  7.3891  X 
1.6487,  and  Iogi0e2-5  is  1.0857. 

Four-Place  Logarithms pages  290-291 

To  find  the  logarithm  of  a  given  number:  For  the  integral 
part  (" characteristic")  of  the  logarithm,  count  to  the 
left  from  units'  place  to  the  first  figure  (other  than  zero) 
of  the  number,  thus : 

7654321:0  0     -1  j-2  -3  -4 

93000000.     or     .000305 

log  93000000  =  7.  +.•••;  log  .000305  =  -  4-.  +  . . .. 
For  the  decimal  part  (" mantissa"),  find  the  first  (two) 
figures  of  the  given  number  in  the  left-hand  column  (N) 
of  the  table  and  the  third  figure  at  the  top  of  another 
column.  The  required  mantissa  will  be  found  in  line 
with  the  first  two  figures  and  in  the  column  headed  by 
the  third.  Consider  the  second  or  third  figure,  if  lacking, 
to  be  zero.  E.  g.,  log  7  =  log  70  =  log  700  =  .8451; 
log  .000023  =  log  .0000230  =  -  5.  +  .3617.  The  man- 
tissa is  always  kep_t  positive,  and  a  logarithm  like  the 
last  is  abbreviated  5.3617  to  save  space.  If  the  number 
has  four  significant  figures  find  the  logarithms  of  the 
next  smaller  and  next  larger  marginal  numbers  and  assume 
that  logarithmic  differences  are  proportional  to  the  corre- 
sponding numerical  differences.  Thus,  1.873  would  be 


280  THEORY  OF  MEASUREMENTS 

located  (on  a  scale)  3  tenths  of  the  way  from  1.87  to  1.88; 
therefore  log  1.873  is  likewise '3/10  of  the  way  from  2718 
to  2742,  namely  2725.  (Three  tenths  of  the  tabular 
difference,  24,  will  be  found  from  the  small  marginal 
tables  to  be  7,  and  2718  +  7  =  2725.  The  approximate 
tabular  difference,  D,  is  given  for  each  line,  so  that  only 
the  final  digits  need  be  subtracted.)  If  the  given  number 
has  5  or  more  significant  figures  a  table  in  which  the 
logarithms  are  stated  to  5  or  more  places  must  be  used. 

To  find  the  number  ("  antilogarithm")  that  corresponds 
to  a  given  logarithm:  If  the  given  logarithm  does  not 
occur  in  the  body  of  the  table  determine  its  position  in 
respect  to  the  next  higher  and  lower  tabular  logarithms 
and  use  proportional  parts  as  before.  E.  g.,  .1345  is 
found  to  be  10/32  of  the  way  from  .1335  to  .1367,  hence 
its  antilogarithm  will  be  1.36J$,  or  1.363.  Notice  that 
10/32  can  be  reduced  to  tenths  and  3/10  to  twenty- 
fourths,  mentally,  by  using  the  small  multiplication 
tables  (PP)  in  the  margin. 

Reciprocals  are  easily  determined  mentally  by  using 
a  table  of  logarithms.  E.  g.,  l/e  =  0.3678.  (Foot-note, 
page  281.) 

Squares page  292 

The  use  of  the  small  table  of  squares  will  be  self- 
evident.  Notice  that  the  square  of  a  number  between 
100  and  110,  say  of  100  +  n  or  107,  consists  of  five  figures 
which  are,  in  order,  1,  2n,  n2,  or  1,  14,  49.  The  square 
of  any  number  between  100  and  200  can  be  found  by  the 
same  process,  " carrying"  mentally.  Thus 

1122  =  1  1732  =  1 

24  146 

144  5329 

12544  29929 


APPENDIX  281 

If  either  2(v*)/n  or  2(vz)/n(n  -  1)  is  located  be- 
tween two  consecutive  numbers  -in  the  third  column, 
(n  -b  l/2)2/(.67449)2,  of  the  same  table,  then  the  value 
of  .67449  A/2  (v2)  /nor  .67449  Vs(v2)/n(n  -  1),  as  the  case 
may  be,  will  be  found  opposite  it  in  the  first  column. 
A  very  rough  mental  calculation  will  prevent  taking  a 
value  which  is  VlO  times  too  large  or  small. 

Constants page  292 

The  characteristics  1  and  2  have  been  replaced  by 
9  and  8  respectively. 

Circular  Functions page  292 

In  the  table  of  circular  functions  the  " radian  value," 
natural  sine,  cosine,  tangent,  and  cotangent  are  given  for 
every  degree  of  the  quadrant  (above  45°  use  the  lower 
and  right-hand  margin),  also  the  logarithmic  sine  and 
cosine.  By  subtracting  the  two  latter  from  each  other 
and  from  zero  any  of  the  six  logarithmic  functions  may 
be  obtained  from  the  table  by  inspection.*  Sines  and 

*  When  two  logarithms  are  to  be  added  or  subtracted  it  will  be 
found  more  convenient,  after  a  little  experience,  to  work  from  left 
to  right  than  the  reverse.  This  is  especially  easy  in  finding  recip- 
rocals by  subtracting  from  zero  (as  in  §68,  no.  7) :  beginning  at  the 
left  subtract  each  figure  from  9,  except  the  last  one,  which  is  to  be 
subtracted  from  10.  For  example, 

log  1  =  0  =1.  9  9  9  9  10 
log  *•  =  0.  4  9  7  1    5 

.'.  log  I/TT  =1.  5  0  2  8    5 

Try  working  the  following  exercises  from  left  to  right.  Before 
each  addition  note  whether  the  next  pair  of  figures  will  add  up  to 
more  than  nine  and  so  give  "  one  to  carry."  If  they  add  up  to 
exactly  nine,  look  a  step  farther  to  the  right,  and  so  on. 

4156528436464436146 
+  3228328943238423856 

In  subtracting  from  left  to  right,  before  setting  down  each  partial 
difference  notice  whether  it  will  need  to  be  decreased  by  unity  on 
account  of  the  figures  that  follow. 


282  THEORY  OF  MEASUREMENTS 

cosines  of  any  intermediate  values  can  safely  be  obtained 
by  interpolation,  and  tangents  up  to  tan  70°.  For  the 
sine,  tangent,  and  numerical  measure  of  a  small  angle 
the  equations  at  the  corners  of  the  table  should  be  used 
as  factors.  E.  g.,  sin  3'  =  3  X  .000290888  =  .000872664. 
For  inverse  " radians"  and  tangents  see  page  285. 


FORMULAE 


Thermometry 

F  =  9C/5+32 
R  =  4C/5 
C  =  5(F-32)/9 
C  =  5R/4 

Logarithms 

loge  x  =  logio  as/logio  e  =  2.3025851  logio  x 
logio  x  =  loge  z/log«  10  =  .4342945  log«  x 
logio  2.3025851 =  .3622157 
logio  .4342945  =  1.6377843 

Constants 

ir  =  3. 141593  =  180° 

e  =  2.718282 

x2  =  9.869604 

V*=  1.772454 

I/T=    .3183099 

V2  =  1.414214 

I'  3  =1.732051 

V5  =  2.236068 

V7  =  2.645751 

>'  10  =  3. 162278 

Mensuration 

triangle:  base,  b;  altitude,  a;  area,  ab/2. 
parallelogram:  base,  b;  altitude,  a;  area,  ab. 
circle:  radius,  r;  circumference,  2irr;  area,  irrz. 
ellipse:  major  axis,  2o;  minor  axis,  26;  area,  irab. 
cylinder:  radiug,  r;  length,  1;  surface,  irrz-\-  2-mrl-\-  irr2;  volume,  irrU. 
cone:  radius,  r;  height,  h;  surface,  irr2  +  irrVr2-{-hz;  volume,  irr2A/3. 
pyramid:  area  of  base,  a;  height,  h;  perimeter  of  base,  p;  slant  height,  s; 
surface,  ps/2;  volume,  ah/3. 

sphere:  radius,  r;  surface,  47ir2;  volume,  /ltrr3/3. 


For  5-formulse  see  page  88. 

For  American  wire  gauge  see  p.  101. 


APPENDIX 


283 


EQUIVALENTS 

The  best  determination  of  the  ratio  of  1  metre  to  1  inch  is  39.37043,  and  this  is 
the  value  generally  adopted  in  scientific  work.  The  legal  relationship,  however, 
is  1  metre  =  39.37079  inches  in  Great  Britain  and  1  inch  =  1  /39.37000  metre  in  the 
United  States,  the  "metre"  being  a  number  of  standard  inches  (36ths  of  the 
Imperial  Standard  Yard)  in  the  former  case,  and  the  "inch"  being  defined  as  £ 
certain  fraction  of  the  standard  metre  (International  Prototype)  in  the  latter. 

The  accepted  ratio  of  1  pound  to  1  kilogram  is  .4535924,  and  the  derived 
equivalents  given  in  the  table  have  been  calculated  from  these  two  ratios  and  the 
accepted  relationship  1  litre  =  1000.027  cm*. 

The  U.  S.  dry  measures  and  the  Imperial  measures  have  been  calculated  from 
the  assumptions  that  1  U.  S.  bu.  is  equal  to  2150.420  cu.  in.,  and  1  Imperial 
gallon  (namely,  the  volume  of  10  av.  Ibs.  of  water  at  62°  F.,  barometer  at  30 
inches,  weighed  in  air  against  brass  weights)  is  equal  to  4.545853  litres. 


Unit 

Equivalent 

Logarithm 

Centimetre 

=  0.3937043  inch 

5951701 

Square  cm               .  . 

=  15531.64  X  (Cd  [red]  15°  760  dry  air) 
=  0.1550030  square  inch 

1912173 
1903402 

Cubic  cm  

=  0.999973  mL 

9999883 

Drachm  
U  S  f    3 

=  16.89407  impl.  HI 
=  16.23116  U.  S.  TTL 
=  3.887936  gm. 
—  3  696593  cm.3 

2277342 
2103495 
5897191 
5678017 

Impl.  f.  3  
Foot 

=  3.551543  cm.3 
-  30  47973  cm 

5504170 
4840111 

Square  ft  
Cubic  ft 

=  929.0138  cm.* 
=  28316.09  cm  3 

9680222 
4520333 

Grain  
Gram 

=  64.79893  mgm. 
=  15.43235  grains 

8115678 
1884322 

Inch  

=  2.539978  cm. 

4048299 

Square  in  
Cubic  in  

=  6.451487  cm.2 
=  16.38663  cm.3 

8096598 
2144897 

Kilogram              

=  2.204622  av.  Ibs. 

3433342 

Kilometre  
Litre  (vol.  of  1000  gm. 
HiO). 

Metre 

=  2.679229  Troy  Ibs. 
=  0.6213767  mi. 
=  1.056716  U.  S.  liquid  qt. 
=  0.9082158  U.  S.  dry  qt. 
=  0.8799239  impl.  qt. 
-  3  280869  ft 

4280097 
7933550 
0239583 
9581891 
9444447 
5159889 

Mile 

=  1.093623  yd. 
—  1609  330  metres. 

0388676 
2066450 

Millilitre  (mL)  

=  1.000027  cm.3 

0000117 

Minim  (m)  U.  S  
Impl. 

=  16.23160  U.  S.  TTL 
=  16.89452  impl.  TTL 
=  0.06160990  cm.3 
—  0  05919238  cm  3 

2103612 
2277459 
7896505 
7722658 

Ounce  av  
Troy  

=  28.34953  gm. 
—  31  10348  gm. 

4525458 
4928090 

U.  S.  f.  5  .  . 

=  29.57275  mL 

4708917 

Impl.  f.  5  
Pound  av. 

1  av.  oz.  of  water  at  62°  F.  =  28.41234  cm.3 
—  453  5924  gm 

4535070 
6566658 

Troy  

=  373.2418  gm 

5719903 

Quart  U.  S.  dry  . 

-  1101  192  cm3 

0418630 

U.  S.  liq 

—  9436  3280  cm  3 

9760417 

—  1  136599  cm  3 

0556074 

Ton  long.  .  . 

—  1016  047  kgm 

0069138 

Short 

—  907  1848  kgm 

9576958 

Metric  (1000  kgm.). 
Yard   ... 

=  2204.622  av.  Ib 
—  91  439208  cm 

3433342 
9611324 

APPROXIMATE  EQUIVALENTS 


25  mm.  =  1  inch  60  mgm.  =  1  gr.  15  TTL      =  1  cm.3 

10  cm.     =  4  in.  15  gr.        =  1  gm.  30  cm.3   =  1  fl.  oz. 

40  in.      =  1  metre  30  gm.       =  1  oz.  1  litre    =  1  quart 

8  km.    =  5  miles  11  Ib.        =5  kgm.  1000  cm.3  =  1  litre 

1000  cm.2  =  1  square  ft.  15  Ib./sq.  in.  =  1  atmo.  =  lkgm./cm.2 


For  slide-rule  equivalents  see  page  97. 


284 


THEORY  OF  MEASUREMENTS 


GREEK  ALPHABET 


Letter 

Used  as  a  symbol  for 

A      a* 

alpha 

Rotation  of  polarized  light;    temperature  coefficient  of  ex- 

pansion; angle. 

B      & 

r    y 

beta 
gamma 

Coefficient  of  expansion;  angle. 
Ratio  of  specific  heats;  angle. 

A      S 

delta 

A  small  quantity  ;  a  finite  difference  (A)  ;  difference  of  ... 

E      e 

epsilon 

2.7183. 

z    f 

zeta 

z  (in  co-ordinates). 

II      T; 

eta 

Viscosity;  efficiency-ratio;  y  (in  co-ordinates). 

e    e 

theta 

Temperature;  angle. 

i     i 

iota 

\—  1;  intensity  of  electric  current. 

K        K 

kappa 

Electrical  conductivity;  magnetic  susceptibility. 

A      X 

lambda 

Wave-length;  latitude. 

M     M 

mu 

Index  of  refraction;  coeff.  of  friction;  .0001  cm.  (MM  —  10~7 

N      v 

nu 

cm.);  permeability. 
Reluctivity  (=  I/M). 

2         £ 

xi 

x  (in  co-ordinates). 

O      of 

omicron 

n    IT 

Pi 

3.1416;  product  of  factors  such  as  .  .  .  (II;  cf.  S). 

P      P 

rho 

Density;  radius. 

S      «rj 

sigma 

Sum  of  terms  such  as  ...  (S)  ;  density  of  air  (<r)  ;  Poisson's 

ratio  (<r). 

T         T 

tau 

Time;  temperature:  torque. 

T      u 

upsilon 

Specific  volume  (=  1/p). 

*    0 

phi 

Angle;  function  of  .  .  .  ;  flux. 

X      x 

chi 

Function  of  

^           l£ 

psi 

Solid  angle. 

»          CO 

omega 

Resistance  (Q);  angular  velocity;  dispersive  power. 

*  Usually  written  with  one  pen-stroke,  somewhat  like  a,  to 
avoid  confusion  with  italic  a. 

t  Not  used  as  a  symbol,  on  account  of  liability  to  confusion 

with  o  and  0. 

SIZE  OP  ERRORS 


CHARACTERISTIC  DEVIATIONS 


.03%  to  .1% "very  small 

.1%  to  .3% "small" 

.3%  to  1% "moderate" 

1%  to  3% "large" 

3%  to  10% "very  large' 


p/m* 
a/m 
slm 

= 

.4769363 
.5641895 
.7071066 

log  p/m 
log  a/m 
log  s/m 

= 

6784603 
7514250 
8494849 

pis 

pja 
a/s 

= 

.6744898 
.8453475 
.7978846 

log  pis 
log  p/a 
log  a/s 

= 

8289754 
9270353 
9019401 

•IP 
alp 
s/a 

= 

1.482602 
1.182944 
1.253314 

log  s/p 
log  alp 
log  s/a 

= 

1710246 
0729647 
0980599 

*  m  ..."  modulus  ",  in  the  equation 


GENERAL  SOURCES  OF  ERROR 


Linear  scales  not  parallel;  lack  of  vertically;  etc. 
Faulty  standardization  of  standards  of  comparison. 
Uneven  or  irregular  subdivision  of  standards. 
Inaccurate  "coincidence"  or  "bisection." 
Faulty  estimation  of  tenths  or  other  subdivisions. 
Parallax  in  reading  scales  or  the  position  of  a  pointer. 
Assumed  zero  of  a  scale. 
Friction  (as  in  a  balance)  and  "play"  or  "back-lash." 


APPENDIX 


285 


DENSITY  OF  WATER 


INVERSE  TANGENTS  AND  CIRCULAR  MEASURE 


1 

density 
(gm.  per 

spec.  grav. 
(:H2O  at 

AO  f~*  \ 

n 

is  the  circular  measure  and 
tangent  of 

n 

is  the  circular 
measure  of 

is  the  tangent 
of 

3 

cm.3) 

4°  C.) 

nooi 

0°  00'  20".  6  =  .000100 

Af\ 

OQO  KKf  f\Kff  f\ 

0°C 

.999841 

.99987 

.UUUJL 

.0002 

41  '.3  =  '.000200 

.4U 
.41 

£&  oo  Uo  .U 
23  29  28  .6 

22  17  37  !l 

.0003 

1  01  .9  =  .000300 

.42 

24  03  51  .2 

22  46  56  .7 

2 

.999941 

.99997 

.0004 

1  22  .5  =  .000400 

.43 

24  38  13  .9 

23  16  03  .7 

4 

.999973 

1.00000 

.0005 

1  43  .1  =  .000500 

.44 

25  12  36  .5 

23  44  58  .2 

6 

.999941 

.99997 

.0006 

2  03  .8  =  .000600 

.45 

25  46  59  .2 

24  13  39  .9 

8 

.999849 

.99988 

.0007 

2  24  .4  =  .000700 

.46 

26  21  21  .8 

24  42  08  .7 

10 

.999700 

.99973 

.0008 
.0009 

2  45  .0  =  .000800 
3  05  .6  =  .000900 

.47 
.48 

26  55  44  .5 
27  30  07  .1 

25  10  24  .7 
25  38  27  .6 

12 

.999498 

.99952 

.001 

0°  03'  26".3  =  .001000 

.49 

28  04  29  .7 

26  06  17  .5 

14 
16 

18 

.999244 
.998943 
.998595 

.99927 
.99897 
.99862 

.002 
.003 
.004 

6  52  .5  •=  .002000 
10  18  .8  =  .003000 
13  45  .1  =  .004000 

.50 
.51 
.52 

28°  38'  52".4 
29  13  15  .1 
29  47  37  .7 

26°  33'  54".2 
27  01  17  .7 
27  28  27  .9 

20 

.998203 

.99823 

.005 
.006 

17  11  .3  =  .005000 
20  37  .6  =  .006000 

.53 
.54 

30  22  00  .4 
30  56  23  .0 

27  55  24  .9 
28  22  08  .6 

22 

.997770 

.99780 

.007 

24  03  .9  =  .007000 

.55 

31  30  45  .6 

28  48  38  .8 

24 

.997296 

.99732 

.008 

27  30  .1  =  .008000 

.56 

32  05  08  .3 

29  14  55  .8 

26 

.996783 

.99681 

.009 

30  56  .4  =  .009000 

.57 

32  39  30  .9 

29  40  59  .3 

28 
30 

.996232 
.995646 

.99626 
.99567 

n 

is  the  circular 
measure  of 

is  the  tangent 
of 

.58 
.59 

c.r\ 

33  13  53  .6 
33  48x16  .2 

QA°  OO'  QQ"  Q 

30  06  49  .4 
30  32  26  .2 
^fl°  VI'  4.Q"  *! 

32 
34 
36 
38 

.99502 
.99437 
.99768 
.99296 

.99505 
.99440 
.99371 
.99299 

.01 
.02 
.03 

.04 

0°  34'  22".6 
1  08  45  .3 
1  43  07  .9 
2  17  30  .6 

0°  34'  22".6 
1  08  44  .7 
1  43  06  .1 
2  17  26  .2 

,oU 
.61 
.62 
.63 
.64 

o'±  ££  oo  .y 
34  57  01  .5 
35  31  24  .2 
36  05  46  .8 
36  40  09  .5 

OU  Ot  *r«7  -O 

31  22  59  .5 
31  47  56  .1 
32  12  39  .3 
32  37  09  .3 

40 

.99221 

.99224 

.05 

2  51  53  .2 

2  51  44  .7 

.65 

37  14  32  .1 

33  01  25  .7 

42 

.99144 

.99147 

.06 
.07 

3  26  15  .9 
4  00  38  .5 

3  26  01  .1 
4  00  15  .0 

.66 
.67 

37  48  54  .8 
38  23  17  .4 

33  25  29  .3 
33  49  19  .5 

44 

.99063 

.99066 

.08 

4  35  01  .2 

4  34  26  .1 

.68 

38  57  40  .1 

34  12  56  .5 

46 

.98979 

.98982 

.09 

5  09  23  .8 

5  08  34  .0 

.69 

39  32  02  .7 

34  36  20  .4 

48 

.98893 

.98896 

.10 

5°  43'  46".5 

5°  42'  38".  1 

.70 

40°  06'  25".4 

34°  59'  31".3 

50 

.98804 

.98807 

.11 

6  18  09  .1 

6  16  38  .3 

.71 

40  40  48  .0 

35  22  29  .1 

52 

.98712 

.98715 

.12 

6  52  31  .8 

6  50  34  .0 

.72 

41  15  10  .7 

35  45  14  .0 

54 

.98618 

.98621 

.13 

7  26  54  .4 

7  24  24  .9 

.73 

41  49  33  .3 

36  07  46  .0 

56 

.98522 

.98525 

.14 

8  01  17  1 

7  58  10  .6 

.74 

42  23  56  .0 

36  30  05  .2 

58 

.98422 

.98425 

.15 

8  35  39  .7 

8  31  50  .8 

.75 

42  58  18  .6 

36  52  11  .6 

60 

.98321 

.98324 

.16 
.17 

9  10  02  .4 
9  44  25  .0 

9  05  25  .0 
9  38  53  .0 

.76 

.77 

43  32  41  .3 
44  07  03  .9 

37  14  05  .4 
37  35  46  .6 

62 

.98217 

.98220 

.18 

10  18  47  .7 

10  12  14  .3 

.78 

44  41  26  .5 

37  57  15  .2 

64 

.98110 

.98113 

.19 

10  53  10  .3 

10  45  28  .7 

.79 

45  15  49  .2 

38  18  31  .5 

66 

.98002 

.98005 

.20 

11°  27'  33".0 

11°  18'  35".  8 

.80 

45°  50*  11".8 

38°  39'  35".3 

68 

.97891 

.97894 

.21 

12  01  55  .6 

11  51  35  .2 

.81 

46  24  34  .5 

39  00  26  .8 

70 

.97778 

.97781 

.22 

12  36  18  .3 

12  24  26  .7 

.82 

46  58  57  .1 

39  21  06  .3 

72 

74 
76 

78 

.97663 
.97545 
.97426 
.97304 

.97666 
.97548 
.97429 
.97307 

.23 
.24 
.25 
.26 
.27 

13  10  40  .9 
13  45  03  .5 
14  19  26  .2 
14  53  48  .8 
15  28  11  .5 

12  57  09  .9 
13  29  44  .6 
14  02  10  .5 
14  34  27  .2 
15  06  34  .5 

.83 
.84 
.85 
.86 

.87 

47  33  19  .8 
48  07  47  .8 
48  42  05  .1 
49  16  27  .7 
49  50  50  .4 

39  41  10  .3 
40  01  48  .9 
40  21  52  .3 
40  41  43  .9 
41  01  23  .8 

80 

.97180 

.97183 

.28 

16  02  34  .1 

15  38  32  .1 

.88 

50  25  13  .0 

41  20  52  .0 

82 

.97054 

.97057 

.29 

16  36  56  .8 

16  10  19  .8 

.89 

50  59  35  .7 

41  40  08  .7 

84 

.96927 

.96930 

.30 

17°  11'  19".4 

16°41'57".3 

.90 

51°  33'  58".3 

41°  59'  14".0 

86 

.96797 

.96800 

.31 

17  45  42  .1 

17  13  24  .4 

.91 

52  08  20  .9 

42  18  07  .9 

88 

.96665 

.96668 

.32 

18  20  04  .7 

17  44  40  .8 

.92 

52  42  43  .6 

42  36  50  .6 

90 

.96531 

.96534 

.33 
.34 

18  54  27  .4 
19  28  50  .0 

18  15  46  .4 
18  46  40  .9 

.93 
.94 

53  17  06  .3 
53  51  28  .9 

42  55  22  .2 
43  13  42  .7 

92 

.96396 

.96399 

.35 

20  03  12  .7 

19  17  24  .2 

.95 

54  25  51  .6 

43  31  52  .3 

94 

.96258 

.96261 

.36 

20  37  35  .3 

19  47  56  .0 

.96 

55  00  14  .2 

43  49  51  .1 

96 

.96119 

.96122 

.37 

21  11  58  .0 

20  18  16  .1 

.97 

55  34  36  .9 

44  07  39  .1 

98 

.95978 

.95981 

.38 

21  46  20  .6 

20  48  24  .4 

.98 

56  08  59  .5 

44  25  16  .6 

100 

.95835 

.95838 

.39 

22  20  43  .3 

21  18  20  .8 

.99 

56  43  22  .2 

44  42  43  .5 

286 


THEORY  OF  MEASUREMENTS 


COMPLETE  SQUARES  UP  TO  9992        4-FiG.  SQUARES  AND  SQUARE  ROOTS 


N 

0 

1234 

5 

6789 

D 

PP 

0 

00 

01   04   09   16 

25 

36   49   64   81 

64   63   62   61   60 

10 

1000 

1020  1040  1061  1082 

1102 

1124  1145  1166  1188 

22 

1  6.4  6.3  6.2  6.1  6.0 

11 

12 
13 
14 

1210 
1440 
1690 
1960 

1232  1254  1277  1300 
1464  1488  1513  1538 
1716174217691796 
1988  2016  2050  2074 

1322 
1562 
1822 
2102 

1346  1369  1392  1416 
1588  1613  1638  1664 
1850  1877  1904  1932 
2132  2161  2190  2220 

24 
26 

28 
30 

2  12.8  12.6  12.4  12.2  12.0 
3  19.2  18.9  18.6  18.3  18.0 
4  25.6  25.2  24.8  24.4  24.0 
5i  32.0  31.5  31.0  30.5  30.0 
6  38.4  37.8  37.2  36.6  36.0 

15 

2250 

2280  2310  2341  2372 

2402 

2434  2465  2496  2528 

32 

7  44.8  44.1  43.4  42.7  42.0 
8  51.2  50.4  49.6  48.8  48.0 

16 

2560 

2592  2624  2657  2690 

2722 

2756  2789  2822  2856 

34 

9  57.6  56.7  55.8  54.9  54.0 

17 

2890 

2924  2958  2993  3028 

3062 

3098  3133  3168  3204 

36 

18 

3340 

3276  3312  3349  3386 

3422 

3460  3497  3533  3572 

38 

59 

58 

57 

56 

55 

54 

19 

3610 

3648  3686  3725  3764 

3802 

3842  3881  3920  3960 

40 

1 

5.9 

5.8 

5.7 

5.6 

5.5 

5.4 

20 

4000 

4040  4080  4121  4162 

4202 

4244  4285  4326  4368 

42 

2 

11.8 

11.6 

11.4 

11.2 

11.0 

10.8 

3 

17.7 

17.4 

17.1 

16.8 

16.5 

16.2 

21 

4410 

4452  4494  4537  4580 

4622 

4666  4709  4752  4796 

44 

4 

23.6 

23.2 

22.8 

22.4 

22.0 

21.6 

22 

4840 

4884  4928  4973  5018 

5062 

5108  5153  5198  5244 

46 

5 

29.5 

29.0 

28.5 

28.0 

27.5 

27.0 

23 

5200 

5336  5382  5429  5476 

5522 

5570  5617  5664  5712 

48 

6 

35.4 

34.8 

34.2 

33.6 

33.0 

32.4 

24 

5700 

5808  5856  5905  5954 

6002 

6052  6101  6150  6200 

50 

7 

41  3 

40.6 

39.9 

39.2 

38.5 

37.8 

8 

47.2 

46.4 

45.6 

44.8 

44.0 

43.2 

25 

6,250 

6300  6350  6401  6452 

6502 

6554  6605  6656  6708 

52 

9 

53.1 

52.2 

51,3]  50.4 

49.5 

48.6 

26 

6700 

6812  6864  6917  6970 

7022 

7076  7129  7182  7236 

54 

27 

7250 

7344  7398  7453  7508 

7562 

7618  7673  7728  7784 

56 

53 

52 

51 

5U 

tv 

48 

28 

7S40 

7896  7952  8009  8066 

8122 

8180  8237  8294  8352 

58 

29 

84/0 

8468  8526  8585  8644 

8702 

8762  8821  8880  8940 

60 

1 

5.3 

5.2 

5.1 

5.0 

4.9 

4.8 

2 

10.6 

10.4 

10.2 

10.0 

9.8 

9.6 

30 

9000 

9060  9120  9181  9242 

9302 

9364  9425  9486  9548 

62 

3 

15.9 

15.6 

15.3 

15.0 

14.7 

14.4 

4 

21.2 

20.8 

20.4 

20.0 

19.6 

19.2 

31 
32 
33 
34 

90/0 
1024 
1059 
1150 

9672  9734  9797  9860 
1030  1037  1043  1050 
1096  1102  1109  1116 
1163  1170  1176  1183 

9922 
1056 
1122 
1190 

9986  1005  1011  1018 
1063  1069  1076  1082 
1129  1136  1142  1149 
1197  1204  1211  1218 

6  4 

7 
7 
7 

5 

6 
7 
8 
9 

26.5 
31.8 
37.1 
42.4 
47.7 

26.0 
31.2 
36.4 
41.6 
46.8 

'25.5 
30.6 
35.7 
40.8 
45.9 

25.0 
30.0 
35.0 
40.0 
45.0 

24.5 
29.4 
34.3 
39.2 
44.1 

24.0 
28.8 
33.6 
38.4 
43.2 

35 

1225 

1232  1239  1246  1253 

1260 

1267  1274  1282  1289 

7 

36 

1250 

1303  1310  1318  1325 

1332 

1340  1347  1354  1362 

7 

47 

46 

45 

44 

43 

42 

37 

1305 

1376  1384  1391  1399 

1406 

1414  1421  1429  1436 

8 

1 

4.7 

4.6 

4.5 

4.4 

4.3 

4.2 

38 

144-4 

1452  1459  1467  1475 

1482 

1490  1498  1505  1513 

8 

2 

9.4 

92 

9  0 

8  8 

8  6 

8  4 

39 

1521 

1529  1537  1544  1552 

1560 

1568  1576  1584  1592 

8 

:i 

14.1 

13'.8 

13^5 

13i2 

12^9 

12^6 

40 

1600 

1608  1616  1624  1632 

1640 

1648  1656  1665  1673 

8 

4 
5 

18.8 
23.5 

18.4 
23.0 

18.0 
22.5 

17.6 
22.0 

17.2 
21.5 

16.8 
21.0 

41. 

1681 

1689  1697  1706  1714 

1722 

1731  1739  1747  1756 

8 

(i 

28.2 

27.6 

27.0 

26.4 

25.8 

25.2 

42 

1704 

1772  1781  1789  1798 

1806 

1815  1823  1832  1840 

9 

7 

32.9 

32.2 

31.5 

30.8 

30.1 

29.4 

43 

1849 

1858  1866  1875  1884 

1892 

1901  1910  1918  1927 

9 

8 

37.6 

36.8 

36.0 

35.2 

34.4 

33.6 

44 

1936 

1945  1954  1962  1971 

1980 

1989  1998  2007  2016 

9 

9 

42.3 

41.4 

40.5 

39.6 

38.7 

37.8 

45 

2026 

2034  2043  2052  2061 

2070 

2079  2088  2098  2107 

9 

41 

40 

39 

38 

37 

36 

46 

2116 

2125213421442153 

2162 

2172  2181  2190  2200 

9 

47 

2209 

2218  2228  2237  2247 

2256 

2266  2275  2285  2294 

10 

1 

4.1 

4.0 

3.9 

3.8 

3.7 

3.6 

48 

2304 

2314  2323  2333  2343 

2352 

2362  2372  2381  2391 

10 

2 

82 

80 

7  8 

7  6 

7.4 

7  2 

49 

2401 

2411242124302440 

2450 

2460  2470  2480  2490 

10 

3 

12^3 

12!o 

11.7 

1L4 

11.1 

10^8 

50 

2500 

2510  2520  2530  2540 

2550 

2560  2570  2581  2591 

10 

4 
5 

16.4 
20.5 

16.0 
20.0 

15.6 
19.5 

15.2 
19.0 

14.8 
18.5 

14.4 
18.0 

51 

260J 

2611  262126322642 

2652 

2663  2673  2683  2694 

10 

0 

24.6 

24.0 

23.4 

22.8 

22.221.6 

52 

2704 

2714  2725  2735  2746 

2756 

2767  2777  2788  2798 

11 

7 

28.7 

28.0 

27.3 

26.6 

25.9  25.2 

53 

2809 

2820283028412852 

2862 

2873  2884  2894  2905 

11 

N 

32.8 

32.0 

31.2 

30.4 

29.6!28.8 

54 

29/0 

2927  2938  2948  2959 

2970 

2981  2992  3003  3014 

11 

9 

36.9 

36.0 

35.1 

34.2 

33.3  32.4 

APPENDIX 


287 


COMPLETE  SQUARES  UP  TO  9992        4-FiG.  SQUARES  AND  SQUARE  ROOTS 


N 

0 

1234 

5 

6789 

D 

PP 

55 

3025 

3036  3047  3058  3069 

3080 

3091  3102  3114  3125 

11 

35 

34 

33 

32 

31 

30 

56 

3136 

3147  3158  3170  3181 

3192 

3204  3215  3226  3238 

11 

1 

3.5 

3.4 

3.3 

3.2 

3.1 

3.0 

57 

3249 

3260  3272  3283  3295 

3306 

3318  3329  3341  3352 

12 

2 

7.0 

6.8 

6.6 

6.4 

6.2 

6.0 

58 

3364 

3376338733993411 

3422 

3434  3446  3457  3469 

12 

3 

10.5 

10.2 

9.9 

9.6 

9.3 

9.0 

59 

3481 

3493  3505  3516  3528 

3540 

3552  3564  3576  3588 

12 

4 

14.0 

13.6 

13.2 

12.8 

12.4 

12.0 

5 

17.5 

17.0 

16.5 

16.0 

15.5 

15.0 

60 

3600 

3612  3624  3636  3648 

3660 

3672  3684  3697  3709 

12 

6 

21.0 

20.4 

19.8 

19.2 

18.6 

18.0 

7 

24.5 

23.8 

23.1 

22.4 

21.7 

21.0 

61 

3721 

3733  3745  3758  3770 

3782 

3795  3807  3819  3832 

12 

8 

28.0 

27.2 

26.4 

25.6 

24.8 

24.0 

62 

3844 

3856  3869  3881  3894 

3906 

3919  3931  3944  3956 

13 

9 

31.5 

30.6 

29.7 

28.8 

27.9 

27.0 

63 

3969 

3982  3994  4007  4020 

4032 

4045  4058  4070  4083 

13 

64 

4096 

4109  4122  4134  4147 

4160 

4173  4186  4199  4212 

13 

29 

28 

27 

26 

25 

24 

65 

4225 

4238  4251  4264  4277 

4290 

4303  4316  4330  4343 

13 

1 

2.9 

2.8 

2.7 

2.6 

2.5 

2.4 

66 

4356 

4369  4382  4396  4409 

4422 

4436  4449  4462  4476 

13 

2 

5.8 

5.6 

5.4 

5.2 

5.0 

4.8 

67 

4489 

4502  4516  4529  4543 

4556 

4570  4583  4597  4610 

14 

3 

8.7 

8.4 

8.1 

7.8 

7.5 

7.2 

68 

4624 

4638  4651  4665  4679 

4692 

4706  4720  4733  4747 

14 

4 

11.6 

11.2 

10.8 

10.4 

10.0 

9.6 

69 

4761 

4775  4789  4802  4816 

4830 

4844  4858  4872  4886 

14 

5 

14.5 

14.0 

13.5 

13.0 

12.5 

12.0 

6 

17.4 

16.8 

16.2 

15.6 

15.0 

14.4 

70 

4900 

4914  4928  4942  4956 

4970 

4984  4998  5013  5027 

14 

7 

20.3 

19.6 

18.9 

18.2 

17.5 

16.8 

8 

23.2 

22.4 

21.6 

20.8 

20.0 

19.2 

71 

504* 

5055  5069  5084  5098 

5112 

5127  5141  5155  5170 

14 

9!  26.1 

25.2 

24.3 

23.4 

22.5 

21.6 

72 

5154 

5198  5213  5227  5242 

5256 

5271  5285  5300  5314 

15 

73 

5329 

5344  5358  5373  5388 

5402 

5417  5432  5446  5461 

15 

74 

5476 

5491  5506  5520  5535 

5550 

5565  5580  5595  5610 

15 

V6 

zz 

zl 

ZU 

iy 

18 

75 

5625 

5640  5655  5670  5685 

•5700 

5715  5730  5746  5761 

15 

1 

2.3 

2.2 

2.1 

2.0 

1.9 

1.8 

2 

4.6 

4.4 

4.2 

4.0 

3.8 

3.6 

76 

5775 

5791  5806  5822  5837 

5852 

5868  5883  5898  5914 

15 

3 

6.9 

6.6 

6.3 

6.0 

5.7 

5.4 

77 
78 
79 

5920 
6054 
6241 

5944596059755991 
6100  6115  6131  6147 
6257  6273  6288  6304 

6006  6022  6037  6053  6068 
6162  6178  6194  6209  6225 
6320  6336  6352  6368  6384 

16 
16 
16 

4 

5 
6 

7 

9.2 
11.5 
13.8 
16.1 

8.8 
11.0 
13.2 
15.4 

8.4 
10.5 
12.6 
14.7 

8.0 
10.0 
12.0 
14.0 

7.6 
9.5 
11.4 
13.3 

7.2 
9.0 
10.8 
12.6 

80 

6400 

6416  6432  6448  6464 

6480 

6496  6512  6529  6545 

16 

8 
9 

18.4 
20.7 

17.6 
19.8 

16.8 
18.9 

16.0 
18.0 

15.2 
17.1 

14.4 
16.2 

81 

6561 

6577  6593  6610  6626 

6642 

6659  6675  6691  6708 

16 

82 

6724 

6740  6757  6773  6790 

6806 

6823  6839  6856  6872 

17 

17 

16 

15 

14 

13 

12 

83 
84 

6889  6906  6922  6939  6956 
7050  J7073  7090  7106  7123 

6972 
7140 

6989  7006  7022  7039 
7157  7174  7191  7208 

17 

17 

1 

1.7 

1.6 

1.5 

1.4 

1.3 

1.2 

2 

3.4 

3  2 

3.0 

2.8 

2.6 

2  4 

85 

7225 

7242  7259  7276  7293 

7310 

7327  7344  7362  7379 

17 

3 

5^1 

4.8 

4.5 

4.2 

3.9 

3.6 

4. 

6  8 

6.4 

6.0 

5.6 

5.2 

4.8 

86 

7396 

7413  7430  7448  7465 

7582 

7500  7517  7534  7552 

17 

5 

8.5 

8.0 

7^5 

7.0 

6^5 

6.0 

87 

7569 

7586  7604  7621  7639 

7656 

7674  7691  7709  7726 

18 

y 

10  2 

9.6 

g'0 

8.4 

7.8 

7.2 

88 

7744 

7762  7779  7797  7815 

7832 

7850  7868  7885  7903 

18 

7 

11.9 

11  2 

10  5 

9.8 

9^1 

8.4 

89- 

7921 

7939  7957  7974  7992 

8010 

8028  8046  8064  8082 

18 

8 

13^6 

12^8 

12^0 

11.2 

10.4 

9.6 

g 

is  a 

11  i 

13  fi 

12  fi 

11  71  10.8 

90 

8100 

8118813681548172 

8190 

8208  8226  8245  8263 

18 

.. 

91 

825* 

8299  8317  8336  8354 

8372 

8391  8409  8427  8446 

18 

1  11 

10 

9 

8 

7 

6 

92 

8464 

8482  8501  8519  8538 

8556 

8575  8593  8612  8630 

19 

j 

93 

8645 

8668  8686  8705  8724 

8742 

8761  8780  8798  8817 

19 

1 

1.1 

1.0 

0.9 

0.8 

0.7 

0.6 

94 

8830 

8855  8874  8892  8911 

8930 

8949  8968  8987  9006 

19 

2 

2.2 

2.0 

1.8 

1.6 

1.4 

1.2 

3 

3.3 

3.0 

2.7 

2.4 

2.1 

1.8 

95 

90,25 

9044  9063  9082  9101 

9120 

9139915891789197 

19 

4 

4.4 

4.0 

3.6 

3.2 

2.8 

2.4 

5 

5.5 

5.0 

4.5 

4.0 

3.5 

3.0 

96 

92/0 

9235  9254  9274  9293 

9312 

9332  9351  9370  9390 

19 

6 

6.6 

6.0 

5.4 

4.8 

4.2 

3.6 

97 

9405 

9428  9448  9467  9487 

9506 

9526  9545  9565  9584 

20 

7 

7.7 

7.0 

6.3 

5.6 

4.9 

4.2 

98 

9604 

9624  9643  9663  9683 

9702 

9722  9742  9761  9781 

20 

8 

8.8 

8.0 

7.2 

6.4 

5.6 

4.8 

99 

9801 

9821  9841  9860  9880 

9900 

9920  9940  9960  9980 

20 

9 

9.9 

9.0 

8.1 

7.2 

6.3 

5.4 

288 


THEORY  OF  MEASUREMENTS 


THE  PROBABILITY  INTEGRAL  y  =  -  /   e~xZdx 

TfJ  0 

The  tabular  value  gives  the  area  under  the  curve  y  =  e—xZ  between  the  ordinates  0  and  x 
in  terms  of  the  total  area  between  x  =  0  and  x  =  +  «.     For  Chauvenet's  criterion  see  p.  234. 


X 

0 

1234 

5 

6789 

D 

0.0 

00000 

01128022560338404511 

05637 

06762  07886  09008  10128 

1118 

0.1 
0.2 
0.3 
0.4 

11246 
22270 
32863 
42839 

12362  13476  14587  15695 
23352  24430  25502  26570 
33891349133592836936 
43797  44747  45689  46623 

16800 
27633 
37938 
47548 

17901  18999  20094  21184 
28690  29742  30788  31828 
38933  39921  40901  41874 
48466  49375  50275  51167 

1086 
1035 
965 

883 

0.5 

52050 

52924  53790  54646  55494 

56332 

57162  57982  58792  59594 

792 

0.6 
0.7 
0.8 
0.9 

60386 
67780 
74210 
79691 

61168619416270563459 
68467  69143  69810  70468 
74800  75381  75952  76514 
80188  80677  81156  81627 

64203 
71116 
77067 
82089 

64938  65663  66378  67084 
71754  72382  73001  73610 
77610  78144  78669  79184 
82542  82987  83423  83851 

696 
600 
507 
419 

1.0 

84270 

84681  85084  85478  85865 

86244 

86614  86977  87333  87680 

340 

1.1 
1.2 
1.3 
1.4 

88020 
91031 
93401 
95229 

88353  88679  88997  89308 
91296  91553  91805  92051 
93606  93807  94002  94191 
95385  95538  95686  95830 

89612 
92290 
94376 
95970 

89910  90200  90484  90761 
92524  92751  92973  93190 
94556  94731  94902  95067 
96105  96237  96365  96490 

270 
211 
162 
121 

1.5 

96611 

96728  96841  96952  97059 

97162 

97263  97360  97455  97546 

89 

1.6 
1.7 
1.8 
1.9 

97635 
98379, 
98909 
99279 

97721  97804  97884  97962 
98441  98500  98558  98613 
98952  98994  99035  99074 
99309  99338  99366  99392 

98038 
98667 
99111 
99418 

98110  98181  98249  98315 
98719  98769  98817  98864 
99147  99182  99216  99248 
99443  99466  99489  99511 

64 
45 
31 
21 

2.0* 

99532 

*55248  57195  59063  60858 

62581 

64235  65822  67344  68805 

1400 

2.1* 
2.2* 
2.3* 
2.4* 

70205 
81372 
88568 
93115 

71548  72836  74070  75253 
82244  83079  83878  84642 
89124  89655  90162  90646 
93462937939410894408 

76386 
85373 
91107 
94694 

77472  78511  79505  80459 
86071  86739  87377  87986 
91548  91968  92369  92751 
94966  95226  95472  95707 

913 

582 
364 
223 

2.5* 

95930 

96143  96345  96537  96720 

96893 

97058  97215  97364  97505 

135 

2.6* 
2.7* 
2.8* 
2.9* 

97640 
98657 
99250 
99589 

97767  97888  98003  98112 
98732  98802  98870  98933 
99293  99334  99372  99409 
99613  99636  99658  99679 

98215 
98994 
99443 
99698 

98313  98406  98494  98578 
99051  99105  99156  99204 
99476  99507  99536  99563 
99716  99733  99750  99765 

79 
46 
26 
14 

3.0* 

99779 

99793  99805  99817  99829 

99839 

99849  99859  99867  99876 

8 

3.1* 
3.2* 
3.3* 
3.4* 

99884 
99940 
99969 
99985 

99891998989990499910 
99944  99947  99951  99954 
99971  99973  99975  99977 
99986  99987  99988  99989 

99916 
99957 
99978 
99989 

99921  99926  99931  99936 
99960  99962  99965  99967 
99980  99981  99982  99984 
99990  99991  99991  99992 

4 
2 
1 
1 

3.5f 

99993 

f99309  99358  99403  99445 

99485 

99521  99555  99587  99617 

27 

3.6f 

3.71 
3.81 
3.91 

99644 
99833 
99923 
99965 

99670996949971699736 
99845  99857  99867  99877 
99929  99934  99939  99944 
99968  99970  99973  99975 

99756 
99886 
99948 
99977 

99773  99790  99805  99819 
99895  99903  99910  99917 
99952  99956  99959  99962 
99979  99980  99982  99983 

14 
6 
3 
2 

4.  t 

99985 

99993  99997  99999  00000 

00000 

00000  00000  00000  00000 

. 

1 

x 
.47694 

y 

0.0 

00000 

0.5 

26407 

1.0 

50000 

1.1 

54188 

1.2 

58171 

1.3 

61942 

1.4 

65498 

1.5 

68833 

1.6 

71949 

1.7 

74847 

1.8 

77528 

1.9 

79999 

2.0 

82266 

2.1 

84335 

2.2 

86216 

2.3 

87918 

2.4 

89450 

2.5 

90825 

2.6 

92051 

2.7 

93141 

2.8 

94105 

2.9 

94954 

3.0 

95698 

3.1 

96346 

3.2 

96910 

3.3 

97397 

3.4 

97817 

3.5 

98176 

3.6 

98482 

3.7 

98743 

3.8 

98962 

3.9 

99147 

4.0* 

*30228 

4.1* 

43137 

4.2* 

53857 

4.3* 

62718 

4.4* 

69998 

4.5* 

75957 

4.6* 

80816 

4.7* 

84759 

4.8* 

87935 

4.9* 

90500 

5.0* 

92549 

*  Beginning  with  x  =  2.01  and  x/,47694  =  4.0  the  (seven-place)  y- values  have  their  first  two 
figures  (9's)  omitted. 

t  Beginning  with  x  =  3.51  the  (nine-place)  y-values  have  their  first  four  figures  (.9999)  omitted. 


APPENDIX 


289 


FIVE-PLACE  LOGARITHMS 


N 

0 

1     234 

5 

6789 

D 

100 

00000 

00043  00087  00130  00173 

00217 

00260  00303  00346  00389 

43 

101 

00432 

00475  00518  00561  00604 

00647 

00689  00732  00775-00817 

43 

102 

00860 

00903  00945  00988  01030 

01072 

01115-01157  01199  01242 

42 

103 

01284 

01326  01368  01410  01452 

01494 

01536  01578  01620  01662 

41 

104 

01703 

01745  01787  01828  01870 

01912 

01953  01995-02036  02078 

41 

105 

02119 

02160  02202  02243  02284 

02325 

02.366  02407  02449  02490 

41 

106 

02531 

02572  02612  02653  02694 

02735- 

02776  02816  02857  02898 

40 

107 

02938 

02979  03019  03060  03100 

03141 

03181  03222  03262  03302 

40 

108 

03342 

03383  03423  03463  03503 

03543 

03583  03623  03663  03703 

40 

109 

03743 

03782  03822  03862  03902 

03941 

03981  04021  04060  04100 

39 

EXPONENTIALS 


FIFTH  PLACE  OF  LOGARITHMS 


X 

e 

ogio  ex 

e~x 

e-2 

.0001 

1.0001 

044343 

.9999 

.0000 

.0002 

1.0002 

Oi8686 

.9998 

.0000 

.0003 

1.0003 

Osl303 

.9997 

.0000 

.0004 

1.0004 

031737 

.9996 

1.0000 

.0005 

1.0005 

Oa2171 

.9995 

1.0000 

.0006 

1.0006 

032606 

.9994 

1.0000 

.0007 

1.0007 

033040 

.9993 

1.0000 

.0008 

1.0008 

033474 

.9992 

1.0000 

.0009 

1.0009 

033909 

.9991 

1.0000 

.001 

1.0010 

034343 

.9990 

1.0000 

.002 

1.0020 

038686 

.9980 

1.0000 

.003 

1.0030 

021303 

.9970 

1.0000 

.004 

1  .0040 

021737 

.9960 

1.0000 

.005 

1.0050 

0?2171 

.9950 

1.0000 

.006 

1.0060 

022606 

.9940 

1.0000 

.007 

1.0070 

023040 

.9930 

1.0000 

.008 

1.0080 

023474 

.9920 

1.0000 

.009 

1.0090 

023909 

.9910 

1.0000 

.01 

1.0100 

024343 

.9900 

.9999 

.02 

1.0202 

028686 

.9802 

.9996 

.03 

1.0305 

Oil303 

.9704 

.9991 

.04 

1.0408 

Oil737 

.9608 

.9984 

.05 

1.0513 

Oi2171 

.9512 

.9975 

.06 

1.0618 

Oi2606 

.9418 

.9964 

.07 

1.0725 

Oi3040 

.9324 

.9951 

.08 

1.0833 

Oi3474 

.9231 

.9936 

.09 

1.0942 

Oi3909 

.9139 

.9919- 

.1 

1.1052 

Oi4343   .9048 

.9900 

.2 

1.2214 

Oi8686 

.8187 

.9608 

.3 

1.3499 

0.1303 

.7408 

.9139 

-    .4 

1.4918 

0.1737 

.6703 

.8521 

.5 

1.6487 

0.2171 

.6065 

.7788 

.6 

1.822 

0.2606 

.5488 

.6977 

.7 

2.0138 

0.3040 

.4966 

.6126 

.8 

2.2255 

0.3474!  .4493 

.5273 

.9 

2.459 

0.3909 

.4066 

.4449 

1. 

2718 

04343 

.3679 

.3679 

2. 

7.389 

0.8686 

.1353 

i!832 

3. 

20.086 

1.3029 

.0498 

31234 

4 

54.598 

1.7372 

.018 

61125 

5. 

148.41 

2.171£ 

.006 

io!389 

6. 

403.43 

2.605£ 

;  .002 

152320 

7. 

1096.6 

3.04011  .000 

2i5196 

8. 

2981.0 

3.4744   .000 

271604 

9. 

8103.1 

3.908" 

r   .000 

368359 

10. 

22029. 

4.342< 

)   .000 

si7203 

N 

0 

1234  5 

6789 

11 

9 

2250  0 

6985 

12 

8 

9512  1 

7019 

13 

4 

7750  3 

4281 

14 

3 

2946  7 

5269 

15 

6 

8-192  3 

2060 

16 

2 

3204  8 

1219 

17 

5 

0355  4 

1725 

18 

7 

5752  7 

1466 

19 

6 

3060  3 

6775 

20 

3 

0503  6 

7765 

21 

2 

8481  4 

5664 

22 

2 

9505  8 

1334 

23 

3 

1962-  1 

1550 

24 

1 

2219  7 

4050 

25 

4 

7023  4 

t  4320 

26 

7 

40(50  I 

»  5135 

27 

6 

7765  £ 

>  1540 

28 

6 

1592  4 

t  7890 

29 

0 

9875  '< 

\  9627 

30 

2 

7147  ( 

)  2456 

31 

fl 

5543  ] 

L  063.9 

32 

1505  i 

?  2570 

33 

1 

3445  i 

t  4320 

34 

8 

5396  '< 

I  8383 

35 

7 

1470  I 

}  5789 

36 

0 

1110  , 

9  8753 

37 

0 

7417  . 

J  9494 

38 

6 

2603  t 

3  0135 

39 

6 

8990 

3  0987 

40 

6 

4318 

3  3962 

41 

8 

4050 

5  9481 

42 

5147 

9  1346 

43 

rt 

8899 

9  9876 

44 

5 

4208 

3  3185 

45 

8406 

1  6271 

46 

Q 

0482 

5  9257 

47 

1 

2468 

9  1234 

48 

i 

5555 

4  4321 

49 

0 

5753 

1  8630 

50 

4073 

9  5162 

51 

2726 

1  5937 

52 

4703 

6  9136 

53 

9134 

5  6789 

54 

0000 

0  9987 

55 

5431 

9  7631 

N  0 

1234 

5 

6789 

56  9 

6415 

5 

551 

57  7 

4051 

7 

535 

58  3 

5271 

6 

3452 

59  6 

0250 

2 

703 

60  6 

7024 

6 

002 

61  3 

4567 

8 

000 

62  {, 

0005 

8 

765 

63  4 

3200 

7 

420 

64  i 

6410 

6 

054 

65  1 

5515 

4 

730 

66  A 

0617 

2 

353 

67  1 

'  2726 

0 

037 

68  ] 

5526 

0 

602 

69  £ 

>  5136 

8 

365 

70  ( 

)  2467 

9 

0235 

71  t 

1  7500 

1 

223 

72  : 

5  4444 

4 

4333 

73  '< 

5  2100 

0 

5764 

74  I 

5  2007 

6 

4205 

75  t 

?  4207 

5 

2074 

76 

L  5520 

6 

3063 

77  t 

t  5284 

0 

6254 

78  t 

t  5162 

7 

2735 

79  ; 

J  5372 

7 

1605 

80  . 

-)  3726 

0 

4715 

81  , 

1  2692 

0 

0255 

82 

L  4703 

fl 

5135 

83  < 

?  2057 

9 

1346 

84 

3  0134 

6 

7501 

85 

2  3456 

7 

7500 

86 

3  0111 

r 

2222 

87 

2  2211 

\ 

0000 

.88 

S  5765 

3210 

89 

9  8654 

c 

1056 

90 

4  2197 

f 

3106 

91 

4  2975 

2 

0742 

92 

9  6307 

t 

1552 

93 

8  5255 

5407 

94 

3  0517 

3 

0517 

95 

2  5405 

0 

6172 

96 

7  2535 

<. 

5352 

97 

7  2716 

( 

5045 

98 

3  715C 

i 

5260 

99 

4  715i 

L 

6037 

10 

0  204C 

1523 

A  subscript  4  means  .0000;  etc. 


Before    annexing   an   italic   figure   subtract    1 
from  the  fourth  decimal  place. 


290 


THEORY  OF  MEASUREMENTS 


FOUR-PLACE  LOGARITHMS 


N 

0 

1234 

5 

6789 

D 

PP 

10 

0000* 

0043*  0086*  0128*  0170* 

0212* 

0253*  0294*  0334*  0374* 

40 

43 

42 

41 

40 

11 

0414 

0453  0492  0531  0569 

0607 

0645  0682  0719  0755 

37 

1 

4.3 

4.2 

4.1 

4.0 

12 

0792 

0828  0864  0899  0934 

0969 

1004  1038  1072  1106 

33 

2 

8.6  8.4 

8.2 

8.0 

13 

1139 

1173  1206  1239  1271 

1303 

1335  1367  1399  1430 

31 

3 

12.9  12.6 

12.3 

12.0 

14 

1461 

1492  1523  1553  1584 

1614 

1644  1673  1703  1732 

29 

4 

17.2  16.8 

16.4 

16.0 

5 

21.5!  21.0 

20.5 

20.0 

15 

1761 

1790  1818  1847  1875 

1903 

1,931  1959  1987  2014 

27 

6 

25.8  25.2  24.6 

24.0 

7 

30.1 

29.4 

28.7 

28.0 

16 

2041 

2068  2095  2122  2148 

2175 

2201  2227  2253  2279 

25 

8 

34.4 

33.6 

32.8 

32.0 

17 

2304 

2330  2355  2380  2405 

2430 

2455  2480  2504  2529 

24 

g 

38.7 

37.8 

36.9 

36.0 

18 

2553 

2577  2601  2625  2648 

2672 

2695  2718  2742  2765 

23 

19 

2788 

2810  2833  2856  2878 

2900 

2923  2945  2967  2989 

21 

39 

38 

37 

36 

20 

3010 

3032  3054  3075  3096 

3118 

3139  3160  3181  3201 

21 

1 

3.9 

3.8 

3.7 

3.6 

7  8 

7  ft 

7  4 

7  2 

21 

3222 

3243  3263  3284  3304 

3324 

3345  3365  3385  3404 

20 

Q 

i  .0 
117 

/  .O 
11  4 

i  .rt 
11  1 

10  8 

22 
23 

24 

3424 
3617 
3802 

3444  3464  3483  3502 
3636  3655  3674  3692 
3820  3838  3856  3874 

3522 
3711 
3892 

3541  3560  3579  3598 
3729  3747  3766  3784 
3909  3927  3945  3962 

19 
18 
17 

2  •"*!  iO  CD 

15^6 
19.5 
23.4 

15.2 
19.0 

22.8 

14.8 
18.5 
22.2 

14^4 
18.0 
21.6 

25 

3979 

3997  4014  4031  4048 

4065 

4082  4099  4116  4133 

17 

7 

27.3 

26.6 

25.9 

25.2 

8 

31.2 

30.4 

29.6 

28.8 

26 

4150 

4166  4183  4200  4216 

4232 

4249  4265  4281  4298 

16 

9 

35.1 

34.2 

33.3 

32.4 

27 

4314 

4330  4346  4362  4378 

4393 

4409  4425  4440  4456 

16 

28 

4472 

4487  4502  4518  4533 

4548 

4564  4579  4594  4609 

15 

29 

4624 

4639  4654  4669  4683 

4698 

4713  4728  4742  4757 

14 

30 

34 

33 

ax 

30 

4771 

4786  4800  4814  4829 

4843 

4857  4871  4886  4900 

14 

1 

2 

3.5 

7.0 

3.4 

6.8 

3.3 

6.6 

3.2 

6.4 

31 
32 

4914 
5051 

4928  4942  4955  4969 
5065  5079  5092  5105 

4983 
5119 

4997  5011  5024  5038 
5132  5145  5159  5172 

13 

13 

3 
4 

10.5 
14.0 

10.2  9.9 
13.6  13.2 

9.6 
12.8 

33 

5185 

5198  5211  5224  5237 

5250 

5263  5276  5289  5302 

13 

5 

17.5 

17.0  16.5 

16.0 

34 

5315 

5328  5340  5353  5366 

5378 

5391  5403  5416  5428 

13 

6 

21.0 

20.4  19.8 

19.2 

7 

24.5 

23.8  23.1 

22.4 

35 

5441 

5453  5465  5478  5490 

5502 

5514  5527  5539  5551 

12 

8 

28.0 

27.2  '  26.4 

256 

36 

5563 

5575  5587  5599  5611 

5623 

5635  5647  5658  5670 

12 

9 

31.5 

30.6|  29.7 

28.8 

37 

5682 

5694  5705  5717  5729 

5740 

5752  5763  5775  5786 

12 

38 

5798 

5809  5821  5832  5843 

5855 

5866  5877  5888  5899 

12 

31 

30 

29 

28 

39 

5911 

5922  5933  5944  5955 

5966 

5977  5988  5999  6010 

11 

1 

31 

3.0 

2.9 

2.8 

40 

6021 

6031  6042  6053  6064 

6075 

6085  6096  6107  6117 

11 

2 

6.2 

6.0 

5.8 

5.6 

3 

9.3 

9.0 

8.7 

8.4 

41 

6128 

6138  6149  6160  6170 

6180 

6191  6201  6212  6222 

10 

4 

12.4 

19  n 

11.6 

11.2 

42 
43 

6232 
6335 

6243  6253  6263  6274 
6345  6355  6365  6375 

6284 
6385 

6294  6304  6314  6325 
6395  6405  6415  6425 

10 
10 

5 

f, 

15.5\  15.0 

18  61  1S  n 

14^5 
17  4 

u!o 

16  8 

44 

6435 

6444  6454  6464  6474 

6484 

6493  6503  6513  6522 

10 

7 

2L7 

21.0 

20^3 

19.6 

45 

6532 

6542  6551  6561  6571 

6580 

6590  6599  6609  6618 

10 

8 
9 

24.8 
27.9 

24.0 
27.0 

23.2 
26.1 

22.4 
25.2 

46 

6628 

6637  6646  6656  6665 

6675 

6684  6693  6702  6712 

9 

47 

6721 

6730  6739  6749  6758 

6767 

6776  6785  6794  6803 

9 

48 

6812 

6821  6830  6839  6848 

6857 

6866  6875  6884  6893 

9 

ZY 

zo 

20 

24 

49 

6902 

6911  6920  6928  6937 

6946 

6955  6964  6972  6981 

9 

1 

2.7 

2.6 

2.5 

2.4 

50 

6990 

6998  7007  7016  7024 

7033 

7042  7050  7059  7067 

g 

2 
3 

5.4 
8.1 

5.2 

7.8 

5.0 

7.5 

4.8 
7.2 

51 

7076 

7084  7093  7101  7110 

7118 

7126  7135  7143  7152 

8 

4 

10.8 

10.4 

10.0 

9.6 

52 

7160 

7168  7177  7185  7193 

7202 

7210  7218  7226  7235 

8 

5 

13.5 

13.0 

12.5 

12.0 

53 

7243 

7251  7259  7267  7275 

7284 

7292  7300  7308  7316 

8 

B 

16.2 

15.6 

15.0 

14.4 

54 

7324 

7332  7340  7348  7356 

7364 

7372  7380  7388  7396 

8 

7 

18.9 

18.2 

17.5 

16.8 

8 

21.6 

20.8 

20.0 

19.2 

55 

7404 

7412  7419  7427  7435 

7443 

7451  7459  7466  7474 

8 

9 

24.3 

23.4 

22.5 

21.6 

*  Interpolated  values  are  given  on  another  page. 


APPENDIX 


291 


FOUR-PLACE  LOGARITHMS 


N 

0 

1234 

5 

6789 

D 

PP 

55 

7404 

7412  7419  7427  7435 

7443 

7451  7459  7466  7474 

8 

23 

22 

21 

20 

56 

7482 

7490  7497  7505  7513 

7520 

7528  7536  7543  7551 

57 

7559 

7566  7574  7582  7589 

7597 

7604  7612  7619  7627 

1 

2.3 

2.2 

2.1 

2.0 

58 

7634 

7642  7649  7657  7664 

7672 

7679  7686  7694  7701 

2 

4.6 

4.4 

4.2 

4.0 

59 

7709 

7716  7723  7731  7738 

7745 

7752  7760  7767  7774 

3 

6.9 

6.6 

6.3 

6.0 

4 

9.2 

8.8 

8.4 

8.0 

60 

7782 

7789  7796  7803  7810 

7818 

7825  7832  7839  7846 

7 

5 

11.5 

11.0 

10.5 

10.0 

6 

13.8 

13.2 

12.6 

12.0 

61 

7853 

7860  7868  7875  7882 

7889 

7896  7903  7910  7917 

7 

16.1 

15.4 

14.7 

14.0 

62 

7924 

7931  7938  7945  7952 

7959 

7966  7973  7980  7987 

8 

18.4 

17.6 

16.8 

16.0 

63 

7993 

8000  8007  8014  8021 

8028 

8035  8041  8048  8055 

9 

20.7 

19.8 

18.9 

18.0 

64 

8062 

8069  8075  8082  8089 

8096 

8102  8109  8116  8122 

65 

8129 

8136  8142  8149  8156 

8162 

8169  8176  8182  8189 

6 

19 

18 

17 

16 

1 

1  n 

1  8 

1  7 

1  6 

66 

8195 

8202  8209  8215  8222 

8228 

8235  8241  8248  8254 

i 
2 

i  -  - 
3  8 

l.O 

3  6 

1.  1 
3  4 

3.2 

67 

8261 

8267  8274  8280  8287 

8293 

8299  8306  8312  8319 

3 

5^7 

5.4 

4^8 

68 

8325 

8331  8338  8344  8351 

8357 

8363  8370  8376  8382 

7  6 

7  2 

6  8 

64 

69 

8388 

8395  8401  8407  8414 

8420 

8426  8432  8439  8445 

5 

9^5 

9^0 

8.5 

s'.o 

g 

11  4 

10  8 

10  2 

9  6 

70 

8451 

8457  8463  8470  8476 

8482 

8488  8494  8500  8506 

7 

7 

13^3 

i2.'e 

1L9 

112 

g 

15  2 

14  4 

13  6 

12.8 

71 

8513 

8519  8525  8531  8537 

8543 

8549  8555  8561  8567 

17.1 

ICO 

i  c'o 

144 

72 

8573 

8579  8585  8591  8597 

8603 

8609  8615  8621  8627 

1D.Z 

1O.O 

i^.*t 

73 

8633 

8639  8645  8651  8657 

8663 

8669  8675  8681  8686 

74 

8692 

8698  8704  8710  8716 

8722 

8727  8733  8739  8745 

15 

14 

13 

12 

75 

8751 

8756  8762  8768  8774 

8779 

8785  8791  8797  8802 

6 

1 

1.5 

1.4 

1.3 

1.2 

2 

3.0 

2.8 

2.6 

2.4 

76 

8808 

8814  8820  8825  8831 

8837 

8842  8848  8854  8859 

3 

4.5 

4.2 

3.9 

3.6 

77 

8865 

8871  8876  8882  8887 

8893 

8899  8904  8910  8915 

4 

6.0 

5.6 

5.2 

4.8 

78 

8921 

8927  8932  8938  8943 

8949 

8954  8960  8965  8971 

5 

7.5 

7.0 

6.5 

6.0 

79 

8976 

8982  8987  8993  8998 

9004 

9009  9015  9020  9025 

6 

9.0 

8.4 

7.8 

7.2 

7 

10.5 

9.8 

9.1 

8.4 

80 

9031 

9036  9042  9047  9053 

9058 

9063  9069  9074  9079 

6 

8 

12.0 

11.2 

10.4 

9.6 

9 

13.5 

12.6 

11.7 

10.8 

81 

9085 

9090  9096  9101  9106 

9112 

9117  9122  9128  9133 

82 

9138 

9143  9149  9154  9159 

9165 

9170  9175  9180  9186 

83 

9191 

9196  9201  9206  9212 

9217 

9222  9227  9232  9238 

11 

10 

9 

8 

84 

9243 

9248  9253  9258  9263 

9269 

9274  9279  9284  9289 

1 

1.1 

1.0 

0.9 

0.8 

85 

9294 

9299  9304  9309  9315 

9320 

9325  9330  9335  9340 

5 

2 
3 

2.2 
3.3 

2.0 
3.0 

1.8 
2.7 

1.6 
2.4 

86 

9345 

9350  9355  9360  9365 

9370 

9375  9380  9385  9390 

4 

4.4 

4.0 

3.6 

3.2 

87 

9395 

9400  9405  9410  9415 

9420 

9425  9430  9435  9440 

5 

5.5 

5.0 

4.5 

4.0 

88 

9445 

9450  9455  9460  9465 

9469 

9474  9479  9484  9489 

6 

6.6 

6.0 

5.4 

4.8 

89 

9494 

9499  9504  9509  9513 

9518 

9523  9528  9533  9538 

7 

7.7 

7.0 

6.3 

5.6 

90 

9542 

9547  9552  9557  9562 

9566 

9571  9576  9581  9586 

4 

8 
9 

8.8 
9.9 

8.0 
9.0 

7.2 
8.1 

6.4 

7.2 

91 

9590 

9595  9600  9605  9609 

9614 

9619  9624  9628  9633 

92 

9638 

9643  9647  9652  9657 

9661 

9666  9671  9675  9680 

•i 

„ 

. 

93 

9685 

9689  9694  9699  9703 

9708 

9713  9717  9722  9727 

94 

9731 

9736  9741  9745  9750 

9754 

9759  9763  9768  9773 

1 

0.7 

0.6 

0.5 

0.4 

95 

9777 

9782  9786  9791  9795 

9800 

9805  9809  9814  9818 

5 

2 
3 

1.4 
2.1 

1.2 
1.8 

1.0 
1.5 

0.8 
1.2 

96 

9823 

9827  9832  9836  9841 

9845 

9850  9854  9859  9863 

4 

2.8 

2.4 

2.0 

1.6 

97 

9868 

9872  9877  9881  9886 

9890 

9894  9899  9903  9908 

5 

3.5 

3  0 

2.5 

2.0 

98 

9912 

9917  9921  9926  9930 

9934 

9939  9943  9948  9952 

6 

4.2 

3.6 

3.0 

2.4 

99 

9956  i  9961  9965  9969  9974 

9978 

9983  9987  9991  9996 

7 

4.9 

4.2 

3.5 

2.8 

8 

5.6 

4.8 

4.0 

3.2 

100 

0000  0004  0009  0013  0017 

0022 

0026  0030  0035  0039 

4 

9 

6.3 

5.4 

4.5 

3.6 

For  natural  logarithms  see  page  114. 


292 


THEORY  OF  MEASUREMENTS 


SQUARES 


CIRCULAR  FUNCTIONS 


n« 


.67449^ 


21762 

24234 

29070 


196 


40061 


52810 
59843 
J'  S°2  67317 

lo   O>i^t  T^OQfi 

19  361  75JdU 

83583 

20  4( 

21  4< 


90       C9Q 

£?     =7A 
24     57o 


26  676 

97      790 

27  29 


12139 


16623 


g  —  17854 
19129 


91011 


61    yt»i 

32  1024 

33  1089 

34  1156  s 

oc    199K    -&blOO 

?fi  Joofi  27702 
36  129     29284 


38  1444 

39  1521 

40  1600 

41  1681 

42  1764 

43  1849 

44  1936 

45  2025 

46  2116 

47  2209 

48  2304 

49  2401 

50  2500 

51  2601 

52  2704 

53  2809 
54 

55 

56  3136 

57  3249 

58  3364 

59  3481 


34296 

36054 
37857 


AKCf\a 

455UO 


53859 
56057 


Tni  cri 

70,16?. 
J2675 

I,*22* 


.674491 


60  3600  I78^ 

61  3721 

62  3844 

co  ^QftQ 

6|  |096  gjjf 

66  |356  94|04 


69     4761 


10314 
10617 


1J237 


13893 


70  4900 

71  f»n4.1 

72  5184 

12200 


77  5929 

78  6084 

79  6241 

80  6400 

81  6561 

82  6724 

83  6889 

84  7056 

85  7225 

86  7396 


89  7921 

90  8100 

91  8281 

92  8464 

93  8649 

94  8836 

96  9216 

97  9409 

98  9604 

99  9801 

100  10000 

101  10201 

102  10404 

103  10609 

104  10816 

105  11025 

106  11236 

107  11449 

108  11664 

109  11881 


17607 


20469 
20896 
21327 

21762 


1^547 
24004 


CONSTANTS 


RAD   DEG   TAN   SIN   g^^OS   C°S  C°T 

i'=io- 

2.9085 

0000    0 

0000  0000  -  oo  Ot   1 

90  * 

01745*  1 
03491  2 
05236  3 
06981  4 

0175  0175  2419  9999  9998  5729 
0349  0349  5428  9997  9994  2864 
0524  0523  7188  9994  9986  1908 
0699  0698  8436  9989  9976  1430 

89  15 
88  15 
87  15 
86  15 

08727  5 

0875  0872  9403  9983  9962  1143 

85  14 

1047   6 
1222   7 
1396   8 
1571   9 

1051  1045  0192  9976  9945  9514 
1228  1219  0859  9968  9925  8144 
1405  1392  1436  9958  9903  7115 
1584  1564  1943  9946  9877  6314 

84  14 
83  14 
82  14 
81  14 

1745  10 

1763  1736  2397  9934  9848  5671 

80  13 

1920  11 
2094  12 
2269  13 
2443  14 

1944  1908  2806  9919  9816  5145 
2126  2079  3179  9904  9781  4705 
2309  2250  3521  9887  9744  4331 
2493  2419  3837  9869  9703  4011 

79  13 

78  13 
77  13 
76  13 

2618  15 

2679  2588  4130  9849  9659  3732 

75  13 

2793  16 
2967  17 
3142  18 
3316  19 

2867  2756  4403  9828  9613  3487 
3057  2924  4659  9806  9563  3271 
3249  3090  4900  9782  9511  3078 
3443  3256  5126  9757  9455  2904 

74  12 
73  12 
72  12 
71  12 

3491  20 

3640  3420  5341  9730  9397  2747 

70  12 

3665  21 
3840  22 
4014  23 
4189  24 

3839  3584  5543  9702  9336  2605 
4040  3746  5736  9672  9272  2475 
4245  3907  5919  9640  9205  2356 
4452  4067  6093  9607  9135  2246 

69  12 
68  11 
67  11 
66  11 

4363  25 

4663  4226  6259  9573  9063  2145 

65  11 

4538  26 
4712  27 
4887  28 
5061  29 

4877  4384  6418  9537  8988  2050 
5095  4540  6570  9499  8910  1963 
5317  4695  6716  9459  8829  1881 
5543  4848  6856  9418  8746  1804 

64  11 
63  11 
62  10 
61  10 

5236  30 

5774  5000  6990  9375  8660  1732 

60  10 

5411  31 
5585  32 
5760  33 
5934  34 

6009  5150  7118  9331  8572  1664 
6249  5299  7242  9284  8480  1600 
6494  5446  7361  9236  8387  1540 
6745  5592  7476  9186  8290  1483 

59  10 
58  10 
57  99 
56  97 

6109  35 

7002  5736  7586  9134  8192  1428 

55  95 

6283  36 
6458  37 
6632  38 
6807  39 

7265  5878  7692  9080  8090  1376 
7536  6018  7795  9023  7986  1327 
7813  6157  7893  8965  7880  1280 
8098  6293  7989  8905  7771  1235 

54  94 
53  92 
52  90 
51  89 

6981  40 

8391  6428  8081  8843  7660  1192 

50  87 

7156  41 
7330  42 
7505  43 
7679  44 

8693  6561  8169  8778  7547  1150 
9004  6691  8255  8711  7431  1111 
9325  6820  8338  8641  7314  1072 
9657  6947  8418  8569  7193  1036 

49  85 
48  83 
47  82 
46  80 

7854  45 

1  7071  8495  8495  7071  1000 

45  78 

i"=io-«x 

4.84814 

COT  COS  JjJjJ  ™°   SIN  TAN  DEO  RA 

SYMBOL       CONSTANT       LOGARITHM 

«•  3.141593  0.4971499 

/I     1800/ir=57°17'45" 

=57°.29578  1.7581226 

=3437'.747  3.5362739 

=206264".80625  5.3144251 

e  2.718282  0.4342945 

M  0.4342945  9.6377843 

*log  T/180  =  log  .01745329  =  8.2418774.     For  sines  and  tangents  of  numerical  angles  see  p.  1 


INDEX 


a  ±d,  211,220 
Abridged  division,  16 

methods,  69 
accuracy  of,  67 

multiplication,  20 
Abscissa,  103 
Abstract  number,  39 
Accidental  errors,  191,  192 
Accuracy,  170 

of  abridged  methods,  67 

of  the  average,  216 

of  calculation,  67 

decimal,  58 

finer  degrees  of,  174 

ideal,  56 

infinite,  171 

and  magnitude,  180 

relative,  63 

in  special  cases,  178 

superfluous,  173 
Addition,  from  left  to  right,  281 

geometric,  90 
Adjustment   of    errors,   (see    least 

squares) 

Advantage  of  dispersion,  214 
Alphabet,  Greek,  284 
American  wire  gauge,  101 
Analysis,  graphic,  128 
Angle(s),  37 

complementary,  48 

as  a  coordinate,  127 

negative,  49 

quadrant  of,  126 

small,  86 

unit  of,  37,  39 
Angular  velocity,  39 
Antilogarithm(s),  74 

with  slide  rule,  101 
Appendix,  tables,  275 
Approximate  calculations,  14 

roots  of  an  equation,  145 
Arbitrary  weighting,  226 
Archimedes'  principle,  223 
Area  by  mensuration,  31 


measurement  of,  30 

unit  of,  27 

Arithmetical  mean,  202 
Assignment  of  weights,  226 
Associative  law,  258 
Assumptions,  143,  148 
Asymmetrical  frequency  distribu- 
tions, 199,  200 
Asymmetry,  201,  210 
Asymptote(s),  139,  148 
Average,  191,  202,  206 

accuracy  of,  216 

deviation,  212,  213 

and  other  deviations,  239 

as  a  least-square  value,  240 

by  partition,  208 

by  symmetry,  207 

weighted,  225 
Axis,  axes, 

change  of,  124 

of  a  graph,  103 

B 

B.  &  S.  wire  gauge,  101 
Balance,  33 

beam  arms,  89 

and  double  weighing,  85 

sensitiveness,  146 
Balancing  columns,  270 
Base,  logarithmic,  71 
Base-line  measurement,  172 
Beam  balance,  33 

double  weighing,  85 

sensitiveness,  146 

unequal  arms,  89 
Bilocular    frequency    distribution, 

199,  200 
Binomial  expansion,  201 

theorem,  10 
Black-thread  method,  131 

and  least  squares,  240 

for  proportionality,  135 
Body  temperature,  110 
Boyle's  law,  137 
Broken  line,  149 


293 


294 


INDEX 


C.G.S.  system,  24 

advantages  of,  39 
Calculation,  accuracy  of,  67 
of  dispersion,  215 
possible  error  after,  176 
probable  error  after,  257,  261 
Caliper,  vernier,  184 
Centimetre,  25 
cubic,  27 
square,  27 

Change  of  graphic  axes,  124 
proportionality  of,  130,  143 
of  scales,  125 
of  variables,  124,  125,  137 
Characteristic,  75,  279 

deviations,  274 
Chauvenet's  criterion,  231 

table  of  values,  234 
Choice  of  graphic  scales,  108 

of  means,  203 
Circle,  elementary,  143 
equation  of,  124,  125 
Circular  functions,  47 

general  definitions,  125 
with  slide  rule,  100 
table  of,  282 
measure,  37 

inverse,  275 
Circumference,  55 
Class  interval,  198 
Classification  of  errors,  191 
Coarse  measurements,  189 
Coefficient  of  expansion,  130 
Coincidence,  measurement  by,  181 
method  of,  180,  182 
principle  of,  180 
Common  logarithms,  71 
Comparison  of  characteristic  de- 
viations, 239 

Complementary  angle(s),  48 
Compound  errors,  257,  261 

units,  15 

Condensed  graphic  scales,  107 
Condition,  135 
Conditioned  measurements,  188 

and  least  squares,  239 
Consecutive  equal  intervals,  245 

249 

Constant(s),  7 
error(s),  191,  192,  271 


by  least  squares,  244 
table  of  282 

Continuity  of  velocity,  156 
Contour,  162 
interval,  165 
lines,  162 

of  earth's  surface,  164 
of  hyperbolic  paraboloid,  164 
map, 

construction,  166 
use,  164 
Convention    of    coordinate    signs, 

158 

Coordinates,  polar,  127 
rectilinear,  127 
in  space,  158 

mathematical,  158 
physical,  158 
Cosecant,  47,  125 
Cosine,  47,  125 
Cotangent,  47,  125 
Criterion,  criteria, 
Chauvenet's,  231 
of  rejection,  230 
of  systematic  errors,  259 
Wright's,  213 
Crude  measurements,  189 
Cube  root  with  slide  rule,  101 
Curve(s)    (see    also    graphic   dia- 
grams) 

drawing  with  ink,  111 
and  equation,  115,  117 
of  errors,  138,  139 
exponential,  138,  139 
interpolation  along,  147 
intersection  of,  127 
logarithmic,  114,  143 
with  new  axes,  124 
probability,  123,  200 
of  sines,  112 
of  tangents,  112 
typical,  136,  138 
Cylinder  points,  100 

D 

d,  211 

di,  dav,  219 

A,  120 

Ay/Ax  =  k,  121,  130 

Danger  in  smoothing  graphs,  111 

Datum,  data, 


INDEX 


295 


insufficient,  148 

plane,  166 
Decimal  (s), 

accuracy,  58 

negative,  253 

places,  65 

written  as  units,  215 
Decimetre  (  =  10  cm.),  44 
Definition,  sharp,  189 
Degree,  37 
Delta  (see  also  difference) 

5,  83 

A,  120 

Ay/Ax  =  k,  130 
Density,  by  Hare's  method,  270 

measurement  of,  34,  39,  223 
direct,  255 

unit  of,  27 

of  water,  133,  285 
Dependent  measurements,  188, 230 
Derived  units,  24 
Deviation(s),  206 

average,  212,  213 

characteristic,  211 
compared,  238,  274 

and  dispersion,  211 

large,  231 

(root-) mean-square,  213 

standard,  212 
Diagonal  scale,  41 
Diagram,  frequency,  197 

graphic,  103  (see  also  graphic) 
Diameters  of  wires,  101 
Difference(s),  in  coordinate  values, 
120,  130 

possible  error  of,  176 

probable  error  of,  257 

relative,  66 

small,  85,  123 

tabular,  76 
Direct  measurements,  188 

of  density,  255 
Directrix  of  a  parabola,  140 
Disagreement    of    measurements, 

189 

Dispersion  (s),  213  (see  also  prob- 
able error) 
of  average  (s),  218 
and  other  characteristic  devia- 
tions, 239 
and  deviation,  211 


fractional,  220 

of  individual  measurements,  219, 
235 

proportional,  220 

relative,  220 

with  slide  rule,  217 

table  of,  216,  292 

and  weighting,  227 

of  zero  value,  220 
Distance,  between  two  points,  140 

from  (xi,  yi)  to  y  =  a  +  bx,  240 
Distortion  of  a  graph,  125 
Distribution,  frequency,  194,  197 
Distributive  law,  260 
Division,  abridged,  20 

with  slide  rule,  95 
Double  position,  145 

weighing,  85 
Drawing  curved  lines,  111 


e,  171  (see  also  constants') 
Earth's  quadrant,  26 

surface,  contours  of,  164 
Element,  of  a  smooth  curve,  143, 

144 
Elementary  circle,  143 

straightness,  144 
Ellipse,  125 

Equal  intervals,  245,  249 
Equation(s),  (see  also  graph) 

with  changed  graphic  axes,  124 

of  a  circle,  124 

of  an  ellipse,  125 

graph  of,  116 

of  a  graph,  123 

interpretation  of,  128 

linear,  129 

intercept  form,  132 

and  locus,  115 

normal,  251 

approximate,  253 

used  as  a  noun,  123 

of  a  parabola,  133 

plotting  of,  118 

polar,  127 

roots  of,  122 

simultaneous,  127 

of  a  standard,  26 

transcendental,  144 

of  two  loci,  149 


296 


INDEX 


Equivalent (s)  for  slide  rule,  97 

table  of,  283 

weights  and  measures,  34,  283 
Error(s),  56,  188 

classification  of,  191 

constant,  191,  192,  271 

curve  of,  138,  139 

of  measurement,  191 

periodic,  265,  266 

possible,  175 

probable,  (see  probable  error) 

progressive,  265,  270 

relative,  64 

size  of,  284 

sources  of,  193,  284 

systematic,  265 
test  for,  265 

and  variations,  194 
Estimation,  measurement  by,  180 

of  tenths,  52 

Expanded  graphic  scales,  107 
Expansion,  binomial,  201 

by  heat,  130 
Exponential  curve,  138,  139 

variation,  154 

Explanatory  notes  (tables),  277 
Extrapolation,  142,  155 

diagram,  154 

F 

F(x),f(x),  146 
False  position,  145 
Figures,  non-significant,  61 

significant,  52,  60,  170 
versus  decimals,  65 

uncertain,  173 

Finer  degrees  of  accuracy,  174 
Finite  differences,  120 
Five-place  logarithm  table,  289 
Fluid  friction,  150 
Focus  of  a  parabola,  140 
Form,  intercept,  132 

standard,  68 
Formulae,  table  of,  282 
Fourth  dimension  of  space,  160 
Fractional  dispersion,  221 

part,  15 

Frequency  diagram,  197,  236 
asymmetrical,  199,  200 
bilocular,  199,  200 
J-shaped,  199,  200 


rectangular,  199,  200 
symmetrical,  199,  200 
U-shaped,  199,  200 
distribution,  194,  197 

types  of,  199 
polygon,  195 
Friction  in  pipes,  150 
Function  (s),  46 
circular,  47 

table  of,  292 
probability,  76 

table  of,  288,  292 
of  small  angles,  86 
Fundamental  units,  24 

G- 

Gauge  points,  slide  rule,  100 
General  sources  of  error,  284 
Geometric  addition,  90 

mean,  202 

Grade,  grade  angle,  22 
Gradient,  22  (see  also  slope) 
Gram,  27 

Graph,  (see  graphic  diagram) 
Graphic  analysis,  128 
axes,  change  of,  124 
determination  of  distance,  140 

of  line  through  two  points,  140 
diagram  (s),  103 

for  Chauvenet's  criterion,  236 

equation  of,  123 

of  an  equation,  116 

of    a    frequency    distribution, 
195 

on  logarithmic  paper,  150,  151 

of  natural  laws,  115 

orientation  .of,  105 

of  a  product,  150 

of  propagated  errors,  263 

of  proportionality,  129 

shifting  of,  124 

of  simultaneous  equations,  127 

slope  of,  109 

smoothing  of,  110 

stretching  of,  125 

of  typical  curves,  138 

a  =  TIT*,  154 

\'a(a  +5)  =  a  +  5/2,  156 

Ax  +  By  +  C  =  0,  121 

ay  =  x\  138,  139 

ax  =  y*,  148 


INDEX 


297 


AT//AX  =  k,  120,  121,  130 

log  y  =  log  a  +  6  log  x,  152 

log  y  =  log  a  +  bx,  153 

me  —  mo  =  2(av  —  me),  204 

p  =  27r>/Z/0~,  154 

0>(3)  =  0,  146 

^>i(x,  2/)<p2(x,  ?/)  =  0,  150 

p  =  I/cos  0,  127 

p  =  2h  sin  0,  127 

p  =  k,  127 

s  =  wo*  +  %aP,  128 

0  =  fc,  127 

x  =  z/2,  148 

x  =  m,  161 

a;  =  a  +  by  +  ey*,  126 

x/a  +  y[b  +z[c  =  I,  168 

x/m  +  y/n  =1,  132 

x2  -  I/2  +  z  =  0,  163 

X2    _|-  ^2    =  C2?   124 

a?  +  2/2  =  0,  127 

X2    _   y2    -    0,    127 

a;2  4.  ^2  4  22  =  52?  162 
f  x2  +  j/2  +  z2  =  52 

I  2    =  fc,   112 

x2  +  (iv  -  fc)2  =  /i2,  127 

(x  -  a)2  +  (y  -  &)2  =  c2,  125 

xy  =  a2,  138,  139 

(x2)(z2)  =  TT  +  log  (sin  x),  157 

y  =  l/x,  y  =  1/x2,  127 

y  =  -  l/x2,  127 

y  =  30/a;,  j/  =  30/x2,  127 

y  =  3  +  23,  128 

y  =  -  l/5x2,  115 

y    =   .99986   +  .0000522x   + 

.0000069x2,  136 
y  =  a  +  6x,    120,     121,    129, 

139,  139 

y  -  a  4-  bx  -f  ex2,  133 
y  =  a  +  bx  +  ex2  +  dx3  •  •  • , 

136 

y  =  a  +  bx  +  k/xz,  157 
y  =  aemxt  y  =  akmx,  153 
y  =  axn,  151,  152 
y  =  o8/3s,  138,  139 
j,  =  e*,  j/  =  em:r,  138,  139 
y  =  e-z2,  123,  138,  139,  232 
y  =  kx,  130 
y  =  log  x,  143 
y  =  m,  161 
j/  =  ip(x),  146 
j/  =  <p(x,  y,  z),  162 


y  =  x,  150 

y  =  x  log  x,  144 

y  =   Vx,  151,  152 

y  =  a;2,  150 

y  =  x2  +  02,  161 

2/  =  z»,  150,  151,  152 

y  =  Z"  -  «,  117,  126 

2/  =  x4,   2/  =  z10,   y  = 

y  =  xm,  150 
y  =  xn,  150 
1/2  =  z(10  -  z)3,  127 
2/2  =  xs,  127 
2  =  w,  161 
z2/9  =  z2  +  3/2,  169 

interpolation,  147 

'representation,  102 

scales,  108 

tables,  147 
Gravity,  specific,  36 

of  water,  table,  285 
Greek  alphabet,  274 


Half,  rounding  off,  29 
Hand  as  a  measure,  29 
Handling  algebraic  decimals,  250 
Hare's  method  (density),  270 
Harmonic  mean,  202,  210 

motion,  156 

Harmony  of  measurements,  189 
Heat,  specific,  235 
Histogram,  195 

modified,  236 
Hooke's  law,  147 
Horizontal  sections,  162,  163 
Hydraulic  friction,  150 
Hyperbola,  127 

rectangular,  138,  139 
Hyperbolic  paraboloid,  163 

section  of  a  surface,  163 


Ideal  accuracy,  56 

Inclination,  22,  42 

Independent  measurements,  188 

Indirect  measurements,  188 
and  graphic  errors,  262 
and  least  squares,  249 
and  probable  errors,  257,  261 

Infinite  accuracy,  171 


298 


INDEX 


Inflection,  point  of,  239 
Inking  curved  lines,  111 
Instantaneous  velocity,  155 
Insufficient  data,  148 
Integral,  probability,  288 
Integrity,  observational,  230 
Intercept  form  of  equation,  132 
International  standards,  26 
Interpolation,  142 

along  a  curve,  graphic,  147 

linear,  143 

logarithmic,  76 

Interpretation  of  equations,  128 
Interquartile  range,  209 
Intersection  of  two  graphs,  127 
Interval  (s),  class,  198 

by  coincidence,  186 

contour,  165 

and  least  squares,  245 
Inverse     tangents     and     circular 

measure,  285 
Irregular  areas,  30 

by  mensuration,  31 

surface,  163 
Irregularities  of  small  groups,  237 


J-shaped    frequency    distribution, 
199,  200 

K 

K.W.H.,  149 
Kilogram,  27 
Kilometre,  26 
Kilowatt-hour,  149 


Large  deviations,  231 
Law(s),  associative,  258 

Boyle's,  137 

of  change,  115 

of  density  of  water,  133 

distributive,  260 

and  graphs,  115 

Hooke's,  147 

linear,  straight  line,  129,  130 
Lead  shot,  specific  heat,  235 
Least  squares,  240 

and  average,  240 

for  black  thread  problem,  240 

for  consecutive  equal  intervals, 
245 


for  a  constant  value,  242 

for  equal  intervals,  245 

for  a  linear  law,  240 

for  a  parabolic  law,  242 

for  proportionality,  243 

for  uniform  intervals,  245 
Left-to-right  addition,  281 
Length,  unit  of,  25 
Light,  velocity  of,  171 
Line,  broken,  149 

initial,  127 

law  of,  130 

straight,  119,  129,  138,  139 
through  origin,  131 
through  two  points,  140 
Linear  equation,  129,  130 
and  black  thread,  131 
intercept  form,  132 

interpolation,  143 

relationship,  137 
Litre,  277 
Locus,  loci,   117  (see  also  graphic 

diagram) 
Logarithm  (s),  base  of,  71 

common,  71 

natural,  71 

properties  of,  71 

for  reciprocals,  79 

with  slide  rule,  101 

tables,  1-place,  72 
3-place,  73 
4-place,  290,  291 
5-place,  289 

Logarithmic  paper,  150,  151 
Lower  quartile,  208 

M 

Magnitude(s)  and  accuracy,  180 

small,  negligible,  80 
Mantissa,  75 
Mass,  unit  of,  27 

true,  85 
Mean(s),  202,  213 

arithmetical,  202 

choice  of,  203 

deviation,  213     • 

geometrical,  202 

harmonic,  202,  210 

quadratic,  202 

relation  between,  204 
Measure(s),  circular,  37 

equivalent,  34 


INDEX 


299 


of  inclination,  42 

numerical,  37 

radian,  87 
Measurement(s),  56 

coarse,  189 

by  coincidence,  181 

conditioned,  135,  188 
and  least  squares,  240 

crude,  189 

of  density,  223 

dependent,  188,  230 

direct,  188 

disagreement  of,  181 

errors  of,  191 

by  estimation,  180 

harmony  of,  189 

independent,  188 

indirect,  188 

and  probable  errors,  257,  261 

possible  error  of,  175 

statement  of,  219 

true  value  of,  191 
Median,  202 

and  average  deviation,  212 

and  weights,  229 
Mental  approximations,  14 
Meter  rates,  149 
Method  of  coincidence,  180,  182 
Metre,  25 
Metric  convention,  25 

equivalents,  283 
Micron,  26 

Microscope,  resolving  power,  171 
Mid-quartile  point,  237 
Milky  way,  171 
Milligram,  27 
Millilitre,  277 
Millimetre,  26 
Minimum  2  («2),  240 
Minute  of  angle,  37 
Mis-named  units,  16 
Mistakes,  191 

Modal  value,  mode,  202,  203 
Model  vernier  caliper,  183 
Modulus,  218 

Motion,  simple  harmonic,  156 
Mound-like  diagram,  198 
Mountain  pass,  164 
Multiple,  probable  error  of,  258 
Multiplication,  abridged,  20 
with  slide  rule,  95 


N 

National  prototype  standards,  25 
Natural  laws  and  graphs,  115 

logarithms,  71 

number,  73 
Negative  angle,  49,  126 

decimals,  253 

and    positive   asymmetry,    201, 

210 

Neutral  web,  25 
Non-significant  figures,  61 
Normal  equations,  251 
approximate,  253 
Numerator  of  zero,  174 
Numerical  measure,  37 

O 

Observations,  weighting  of,  223 
Observational  integrity,  230 
Occasional  large  deviations,  231 
Orbital  period,  150 
Ordinate(s),  103 

summation  of,  112,  133 
Orientation  of  a  graph,  105 
Origin,  103 

change  of,  124 

meaning  of  line  through,  131 
Overlining  negative  decimals,  253 


7T ,  angle,  39 

ratio,  55,  56,  171 
7r-disc  and  least  squares,  248 
*>(*),  146 
^(x,  y)   =  0,  150 
pt)-variation,  137 
Paper,  logarithmic,  150,  151 

semi-logarithmic,  152,  153 
Parabola,  122 

arch-shaped,  122 

festoon-shaped,  122 

finding  equation  for,  133,  148 

semi-cubical,  127 

x  =  2/2,  148 
Vx  +  Vl/  =   V4,  112 

y  =  a  +  bx  +  ex*,  122 

y  =   -  kx2,  116 

y  =  mx*,  116,  122,  124 
Parabolic  section,  163 
Paraboloid,  163 


300 


INDEX 


Parameter,  8 

Partition,  average  by,  208 

Per,  15 

Percent  slope,  23 

Period  of  rotation,  150 

Periodic  errors,  265,  266 

Physical  arithmetic,  13 

measurement,  56 
Places,  decimal,  65 
Plane,  datum,  166 
Planimeter  and  least  squares,  249 
Plotting,  principles  of,  109,  118 

tabular  values,  109 
Point(s),    distance    between    two, 

140 

of  inflection,  239 
line  through  two,  140 
mid-quartile,  237 
Pole,  127 
Polar  coordinates,  127 

equation,  127 
Polygon,  frequency,  195 
Population  and  extrapolation,  55 
Position,  double,  or  false,  145 
Positive  and  negative  asymmetry, 

201,  210 

Possible  error,  175 
Power,  probable  error  of,  260 

resolving,  171 
Prejudice,  54,  230 
Principle  of  Archimedes,  223 

of  coincidence,  180 
Probability    function,    calculation 

of,  77 

curve  of,  123,  200 
integral,  table  of,  288 
Probable   error(s),    117,    235    (see 

also  dispersion) 
and  other  characteristic  devia- 
tions, 239 

of  indirect  measurements,  261 
associative  law,  258 
of  a  difference,  257 
distributive  law,  260 
formulae  for,  261 
of  a  multiple,  258 
of  a  power,  260 
of  a  product,  259 
of  a  quotient,  260 
of  a  sum,  257 
table  of,  292 


Product,  possible  error  of,  176 

probable  error  of,  259 

of  two  equations,  150 
Progressive  errors,  265,  270 
Propagation  of  errors,  255 
Properties  of  deltas,  83 

of  logarithms,  71 

Proportion  with  black  thread,  135 
and  least  squares,  243 

with  slide  rule,  96 
Proportional  changes,  130,  143 

dispersion,  221 
Proportionality,  6 

in  a  graph,  129 
Prototype  standards,  25,  191 
Protractor,  40 


Quadrant,  26 

of  an  angle,  127 
Quadratic  mean,  202 
Quantities,  real  and  ideal,  57 
Quartiles,  upper  and  lower,  208 
Quotient,  possible  error  of,  177 

probable  error  of,  260 

R 

r,  P,  127 

r.m.s.,  (see  root-mean-square) 

r*  =  z2  +  yi,  127 

Radian  measure,  87 

Radius,  127 

Range,  interquartile,  209 

semi-interquartile,  209 

total,  211 

Rates  (of  charge),  146,  149 
Ratio(s),  slide  rule,  96,  186 
Readiness-to-serve  charge,  146 
Reciprocals  with  logarithms,  79 

with  slide  rule,  97 
Rectangular    frequency    distribu- 
tion, 199,  200 

hyperbola,  127,  138,  139 
Redetermination,  230 
Rejection  of  a  measurement,  230 
Relationship,  linear,  137 

between  means,  204 
Relative  accuracy,  63 

difference,  66 

dispersion,  220 


INDEX 


301 


errors,  64 

Repeated  values  and  weights,  225 
Representative  magnitude(s),  201 
average  as  best,  206 
mid-quartile  point,  237 
Resolving  power  of  a  microscope, 

171 
Root(s),  cube,  101 

by  double  position,  145 

-mean-square  deviation,  213 

with  slide  rule,  98 

square,  table,  286,  287 

of  y  =  f(x),  145 
Rotational  period,  150 
Rounding  off,  28,  29,  59 
Rule,  slide,  90 

for  accuracy  of  average,  216 


S,  218 

Eva,  =  0,  207 
S(z>2),  213,  240 
Saddleback,  163,  164 
Scale(s),  diagonal,  41 

graphic,  change  of,  125 
choice  of,  108 
condensed,  107 
expanded,  107 

logarithmic,  90 

reading,  93 
Secant,  47,  125 
Second,  of  angle,  37 

of  time,  28 
Sections,  plane,  162,  163 

hyperbolic,  163 

parabolic,  163 
Semi-cubical  parabola,  127 
Semi-interquartile  range,  209 

and  dispersion,  214 
Semi-logarithmic  paper,  152,  153 
Sensitiveness  and  weighing,  146 
Sharp  definition,  189 
Shifting  a  graphic  curve,  124 
Sigma  notation,  218 
Significant  figures,  52,  60,  170 
versus  decimal  places,  65 

zero,  59,  61,  220« 

Significance  of  the  dispersion,  213 
Signs     of     three-dimensional     co- 
ordinates, 158 


Simple  harmonic  motion,  156 
Simultaneous  equations,  127 

indirect  measurements,  249 
Sine(s),  23,  44,  125 

curve,  112 

table,  292 
Sinusoid,  112 
Size  of  errors,  284 
Skewedness,  201 
Slide  rule,  90 

for  dispersions,  217 
ratios,  186 
Slope,  22 

of  a  graph,  109 

of  a  straight  line,  120 
Small  angle,  86 

difference,  85,  143 

groups  of  measurements,  237 

magnitudes,  80 
Smooth(ed)  curve,  143,  144 
Smoothing  a  curve,  110 

with  black  thread,  131 
Solution  by  double  position,  145 
Sources  of  error,  193,  284 
Space  of  four  dimensions,  160 

of  three  dimensions,  158 
Specific  gravity,  36 
of  water,  285 

heat,  235 

volume,  36 

Sphere,  sections  of,  162 
Squares,  least,  (see  least  squares) 

and  roots,  slide  rule,  98 
tables,  286,  287,  292 
Standard(s)  deviation,  212 
and  other  deviations,  239 

form,  68 

of  measurement,  25,  180 
Statement  of  a  measurement,  219 
Statistics,  194,  197 
Straight  line,  119,  138,  139 
law,  129,  130 
slope  of,  120 
through  two  points,  140 
Straightness,  elementary,  144 
Strained  curve,  125 
Stretching  a  graph,  125 
Substantive  use  of  an  equation,  123 
Substitution,  147 

of  x  +  p,  y  +  q,  124,  126 

of  mx,  x/a,  ny,  y/b,  125,  126 


302 


INDEX 


Subtraction,  left  to  right,  281 

Sum,  possible  error  of,  176 
probable  error  of,  257 

Summation  of  ordinates,  112,  133 

Superfluous  accuracy,  173 

Surface,  irregular,  163 

Surveyor's  base  line,  accuracy  of, 
172 

Symmetrical    frequency    distribu- 
tion, 199,  200 

Symmetry,  average  by,  207 
of  a  curve,  123 

Systematic  errors,  265 
test  for,  259 


6,  127 
Tables,  275 

characteristic  deviations,  284 

Chauvenet's  criterion,  234 

circular  functions,  292 

constants,  292  (see  also  ir) 

6-formula3,  88 

density  of  water,  (134),  275 

equivalents,  283 

errors  classified,  191 
size  of,  284 

exponential  functions,  288,  289 

formulae,  282 

general  sources  of  error,  284 

Greek  alphabet,  284 

logarithms,  1-place,  72 
3-place,  73 
4-place,  290,  291 
5-place,  289 

inverse  tangents,  285 

inverse  circular  measure,  285 

natural  logarithms,  114 

probability  integral,  288 

probable  errors  (indirect),  261 

sines  of  numerical  angles,   113, 
114 

slide-rule  equivalents,  97 

squares,  286,  287,  292 

tangents  of  numerical  angles,  113 

wire  gauge,  101 
Tabular  difference,  76 

values  and  graphs,  109 
Tangent(s),  22,  42,  125 

curve  of,  112 

inverse  (table),  285 


table  of,  292 

tan  6  =  y/x,  127 
Target  diagram,  193 
Temperature  of  the  body,  110 
Tenths,  estimation  of,  52 
Test  for  systematic  errors,  265 
Thread  method  of  smoothing,  131 
Three-dimensional  space,  158 
Time,  28 

intervals  by  coincidence,  186 
Total  range,  211 
Transcendental  equations,  144 
Transformation    to    linear    equa- 
tion, 137 

Tresca  cross-section,  28 
True  value  of  a  measurement,  191 
Two  points  on  a  graph,  distance, 

140 
line  determined  by,  140 

loci,  single  equation  of,  149 
Typical  curves,  136,  138 

(see  also  line,  parabola,  etc.) 
Types   of   frequency   distribution, 

199 

U 
U-shaped   frequency   distribution. 

199,  200 

Uncertain  figures,  173 
Uniform  intervals,  245,  249 
Unit(s),  of  angle,  37,  39 

of  area,  27 

compound,  15 

versus  decimals,  215 

of  density,  27 

derived,  24 

fundamental,  24 

of  length,  25 

of  mass,  27 

mis-named,  16 

of  time,  28 

of  volume,  27 

of  weight,  27 
Upper  quartile,  208 


v,  206,  211 

Values,    physical   and   mathemat- 
ical, 82 

Variable(s),  8,  47 

change  of,  124,  125,  127 


INDEX 


303 


velocity,  155 
Variates,  194 

measurement  of,  194 
Variation  (s),  8 

of  density  (water),  133 

and  errors,  194 

of  measurements,  206 

pv-,  137 

Vector  sum,  192 
Velocity,  angular,  39 

continuity  of,  156 

by  extrapolation,  155 

of  light,  171 

at  a  point,  155 

variable,  155 
Vernier,  183 

caliper,  184 
Volume,  irregular,  32 

specific,  36 

unit  of,  27 

of  water,  27,  285 

W 

Water,  density  of,  133 
and  temperature,  285 

mass  and  volume  of,  27 

specific  gravity  of,  285 
Weighing,  double,  85 

and  sensitiveness,  146 
Weight(s),  arbitrary,  226 

and  dispersion,  227 

and  measures,  34 


for  repeated  values,  225 
Weighted  average,  225 
Weighting  of  observations,  223 
Wire  gauge,  101 
Wright's  criterion,  238 


x  =  k,  graph  of,  121 
x-axis,  103 
z-intercept,  132 
a?2/-plane,  zz-plane,  158 
f  x  =  p  cos  6 
\y  =  p  sin6,  127 
x2  +  2/2  =  c2,  124 
(x  -  a)2  +  (y  -  6)2  =  c2,  125 


y  =  a  +  bx,  120,  121 
\y=P  sin  6 
\x  =  p  cos  6,  127 
2/-axis,  103 
^/-intercept,  132 
2/z-plane,  158 

Z 

z-axis,  158 

Zero  as  a  dispersion,  220 

as  a  fraction,  174 

numerator,  174 

of  a  scale,  55 

significant,  220 


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